First off, I would not recommend any of these telescopes. Of the ones you listed, the 114EQ is the least bad choice. But, due to the poor quality of its mount, it's the better of the bunch.
Ok, before I get into the nuts and bolts, let me first offer a couple pieces of advice:
1.) Join an astronomy club or society. In the US, membership usually runs $50 or less per year. Most clubs offer guidance and advice to new members. They also often conduct frequent star parties. These are events where those who have them bring out telescopes and everyone spends the evening observing together. At such events, you can see a lot of different types of telescopes of all different designs, prices, manufacturers, etc... and talk to owners about them. Clubs also have other programs that can help those new to the hobby learn and better enjoy their observing time.
2.) The best telescopes I can recommend in the under $200 price range (which is where these seem to be) are the AWB OneSky 130 (also known as the SkyWatcher Heritage 130P outside of the US) or the Meade Lightbridge Mini 130. Both are 130 mm Newtonian telescopes with fairly good optics for their price range and stable mounts. Meade makes a 114 mm version, and so does Orion. I would not recommend any telescopes smaller than that, as there's little they can show you that a set of binoculars in the price range wouldn't do just as well or better.
Now, your initial question was about the differences between them. Please bear with me, I tend to be verbose and try to be thorough (in other words, grab a snack and settle in for the read).
First of all, let me clear up a common misconception many people who are new to amateur astronomy tend to have. The primary purpose of an astronomical telescope is not to magnify, but to gather light. The problem with astronomical objects is less one of apparent size (this is a secondary concern, and not unimportant, but not as critical) and more one of brightness. For example, the Andromeda Galaxy, also known as object 31 in the Messier catalog (M31), has an an angular size of about 3° by 1.5°. The full moon is about 1/2° wide - so M31 is a lot bigger in apparent size. But it's very faint in comparison. If you know where to look and have fairly dark skies, you can actually see it with your naked eye at times, but not very clearly.
All the light you see comes into your eye through your pupil. The pupil of an average adult eye, when fully dark adapted, has a diameter of about 6 or so mm. That's about 28 or so square mm of light gathering area. A modest 80mm telescope (80mm being the aperture) has an area of over 5,000 square mm - over 150 times as much light gathering area. All that light is then funneled down and concentrated on your eye. This makes dim objects much brighter. Magnification helps, but it doesn't matter how big an object appears if you can't see it at all.
The light gathering ability of the telescope is dictated by its aperture - the size of the main optical element collecting light (either an objective lens or primary mirror). As such, it's not really adjustable (well, you can adjust it downward by blocking the aperture, but you can't increase it). Magnification, however, IS adjustable. The magnification power of a telescope is calculated by dividing the focal length of the telescope by that of the eyepiece. For example, if you have a telescope with a focal length of 1,000 mm and an eyepiece of 25 mm, you get 40x magnification. If you swap out the eyepiece for a 10 mm eyepiece, you get 100x. Theoretically, you can magnify as much as you like, though you can only magnify so much before the image is a blurry mess (this is also a matter of aperture - larger apertures allow for higher magnification before blurriness sets in).
Thus, the most important consideration when choosing an astronomical telescope is usually the aperture of the telescope.
When most people who are inexperienced with telescopes hear the word telescope, they think of a refractor. A refractor, or refracting telescope, uses a lens to collect light. This lens is known as the objective lens (though sometimes referred to, erroneously, as the primary lens). When light is passed through a substance, such as glass, it is slowed slightly by the substance. An objective lens is shaped with a convex (outward) curvature. When light passes through the thicker, center part of the lens it is slowed more than that in the thinner, outer portion of the lens, This causes the light to refract, or bend. The symmetry of the lens causes the light to all bend toward a point along an axis through the center of the lens. The point where the light rays meet is known as the focal point. All the light it collects is concentrated here. The eyepiece sits here to gather the light and reconstruct the image for viewing.
Refractors were the first telescopes made, invented in the early 1600's in the Netherlands. People often mistakenly consider Galileo the inventor of the telescope. In fact, the man most frequently credited with the invention is Hans Lippershey (though two other men sometimes are given credit). Galileo was only the first to use it to examine the night sky.
Even in his day, however, there was a problem with the design. When you refract light, you cause the colors to separate. This is exactly how a prism works. In fact, in a real sense, a lens IS a prism. This is because different wavelengths of light, what we percieve as colors, are slowed at different rates. Shorter wavelengths, which are bluer light, are slowed more than longer, redder, wavelengths. What this means in a telescope is that the actual focal points of different colors of light are separated with bluer light coming to a focus a bit closer in than redder light.
We call this phenomenon chromatic aberration, or CA, and it can be seen in telescopes as ablueish blur or fringe around objects you're viewing. It degrades the quality of the image.
There are two main methods for reducing its effects. The first is to increase the focal length of the telescope. A lens with a longer focal length has less pronounced curve: the difference between the thinnest and thickest parts of the lens are less dramatic than in a lens with a shorter focal length. This means there's less dramatic bending, which leads to the focal points of the colors being closer together. If you look at early telescopes, like those used by Galileo, they tend to be very long indeed with high focal ratios. The focal ratio is the ratio of the focal length to the aperture. My 8" (203 mm) Schmidt-Cassegrain Telescope (SCT) has a focal length of 2,030 mm, making its focal ratio f/10. My 8" imaging Newtonian has a focal length of only 800 mm, making its focal ratio f/3.9. I believe some of Galileo's early scopes had ratios like f/20 - the tube was 20 times longer than it was wide. Mostly this was to reduce chromatic aberration (though it was also to increase magnification)
The other method of reducing CA is by using multiple lenses with different shapes and made of different glass materials. The most common option is to use two complementary lenses that work together to shift the focal points of bluer and redder light to bring them closer together. This is called a doublet or achromat. Using three lenses makes a telescope a triplet and and is often called an apochromatic refractor. Sometimes even more lenses are used. But this comes at a price. Each lens modifies the light, and while correcting, can also introduce distortion. It also requires 2 surfaces (front and back) per lens to be precisely configured with a specific curvature. This is costly and time consuming. Lastly, lenses are heavy, and with a refractor, they are mounted at the front, which puts the weight at the front, and makes larger and longer such scopes much more difficult to properly maneuver. The largest refractor ever made was only about 49 inches, and the largest used for research was only 40 inches.
Galileo and his contemporaries understood the problem and that it could be solved by using a mirror instead of a lens. Unfortunately, in those days, mirrors were much more difficult to make. They were made from solid metal, which is much more difficult to precisely form and polish. Most often,they were made of a metal known as speculum metal, which tarnishes very easy. It wasn't until Sir Isaac Newton came along about 60 years later that a reflecting telescope was first created.
Newton used a mirror with a parabolic curvature to gather light. It sat at the end of an open-ended tube. Light gathered by this mirror was bent and directed toward a secondary mirror, which, mounted at a 45 degree angle, would send the light out the side of the tube to the waiting eyepiece.
Because a mirror of this kind is all metal, there's no glass to pass through, so the glass does not separate the colors of light. This eliminated chromatic aberration (well, there was still some in the eyepiece, but not much). But, again, mirror technology was a problem. Still, a few people adopted the design, and even built very large reflecting telescopes. William Herschel's telescope was about 48 inches in diameter. It was much easier to maneuver such a telescope, as the weight was all at the rear of the instrument (not to say it was easy, but putting a 4 foot glass lens at the end of the 40 foot tube would have been exceedingly difficult to work with - very, very top-heavy). Herschel's mirrors, however, tarnished easily. He actually had multiple mirrors, so one was being polished while another was in use, that he could just swap out as needed.
In the mid 19th century, the process of depositing a thin layer of metal, originally silver, but now usually aluminum, onto glass was developed. Glass, while heavy, is much easier to configure into a precise shape. Once done, the reflective coating could be deposited onto the glass mirror blank. Because the coating is on the front-side of the mirror, light doesn't pass through the glass, just bounces off the coating. This made Newtonian reflecting telescopes much easier to produce. Only the one surface, the primary mirror, has to be precisely configured, and this made them more cost-effective. And the weight and balance issues made making larger and larger telescopes easier.
Newton's design wasn't the only. Laurent Cassegrain and James Gregory both designed, but probably never constructed, reflecting telescopes relying on mirrors with a hole in the center. The light is collected by the primary, sent to the secondary, and sent back toward the primary to pass through the hole. In Cassegrain's design, the secondary is flat. In Gregory's, it was curved. Both allowed for a shortening of the overall physical length of the instrument.
In more recent times, the use of mirrors and lenses together lead to the development of the Catadioptric telescope. The most common such designs are the Schmidt-Cassegrain Telescope (SCT) and Maksutov-Cassegrain Telescope (Mak).
Just as refractors suffer from Chromatic Aberration, Newtonian telescopes suffer from an aberration called Coma. Coma appears as distortions of the stars around the edge of the field of view. They appear to lengthen and have comet-like tails (hense the name, coma). As with Refractors, the effect is reduced by increasing focal length. In modern telescopes, corrective lenses are often used.
Catadioptric telescopes resolve this partly by using a different kind of mirror shape, a spherical mirror. The curvature of the parabolic mirror used in a Newtonian design varies along its profile. The curvature of a spherical mirror is a constant curve. This makes them easier to manufacture and reduces or eliminates coma, but it introduces another problem: spherical aberration. Catadioptrics overcome this by using the combination of lenses and mirrors to correct the distortions. However, the numerous surfaces then require precise configuring, which increases the cost to manufacture.
Each of these designs has its pros and cons. There isn't one "all-purpose" option that's best for everything.
As far as amateur telescopes go, however, the Newtonian design tends to be the most popular. This is largely due to the average cost per inch of aperture. For example, an 8" Newtonian can be found in the neighborhood fo $300 USD for the telescope tube alone. An 8" SCT tube alone is likely to cost about three or four times that. Very few 8" refractors are available, and those that are would likely cost several thousand dollars.
Now, as to the telescope you chose. The Celestron PowerSeeker 127EQ is actually not a true Newtonian. It is what we call a Bird-Jones (or Jones-Bird, depending on who you talk to) Newtonian. A true Newtonian has a parabolic primary mirror and a flat secondary mirror. The only other optical surfaces are in any eyepieces used. The Bird-Jones Newtonian uses a spherical primary mirror. To counter the spherical aberration inherent in the design, a small correcting lens, which is not much more than a Barlow lens, is included in the optical path (usually mounted inside the focuser mechanism). This allows for a more compact design (the PowerSeeker 127 is only 508 mm (20 inches) long, but has a focal length of 1,000 mm (almost 40 inches). But it also decreases the quality of the view. The optics on this telescope are pretty poor. Worse still, it is extremely difficult to collimate.
All telescopes require the optical elements to be properly aligned to produce the best image. This is collmation. In the case of refractors, it rarely needs adjustment. Most SCT's require it only occasionally, and Maksutovs rarely. But Newtonians require it frequently. Newtonians will have a set of adjustment screws or knobs on the back, behind the primary mirror, and on the front as part of the support that holds the secondary mirror (usually referred to as the spider). Most telescopes use a 3-screw system for adjusting the tilt of the mirrors. It can be a chore at first, but with a little practice, it becomes fast and easy to do. It also requires a tool to insert into the focuser. Traditionally, this is a collimation cap or Cheshire, which work like an eyepiece, but do not have lenses. They help you visualize the alignment of the optics so that you can adjust it properly. You can also get a laser collimator which uses a low-power laser to guide you. But none of these work with a Bird-Jones scope. The correcting lens makes them very, very difficult to collimate.
Furthermore, the mount on the PowerSeeker line, particularly the 127EQ, is not very stable. It wobbles and shakes and makes for difficult observing.
Sadly, the reviews on sites like Amazon make it out to be a great instrument. But these reviews themselves are flawed. Amazon's servers use review rankings to determine the display order of search results. If an item has higher ratings in its reviews, it appears toward the top of the list of results when you search. Most buyers don't look past the first page, and rarely much further than 2 pages. So if a seller wants to sell his or her product, it needs to be on the first page, preferably right at the top. This leads a lot of companies and individual sellers to push customers to add reviews. Some companies also falsify reviews (though it would be difficult to prove it) in order to game the system. As such, the reviews on Amazon (and many other sites) are nearly useless to the consumer. And the PowerSeeker telescopes are a prime example.
On multiple other sites I spend time on, this telescope is outright hated. It is sold by Celestron (a chinese-owned company) to unsuspecting buyers. In the long run, I think it causes more harm to amateur astronomy than good - because users often end up discoraged by its poor peformance and cheap construction.
If you've already purchased it, I'd strongly recommend returning it and getting something different (e.g. the two scopes I listed at the beginning). If you haven't yet, then please do not. The two telescopes I listed are significantly better in nearly every way for a similar price. Even the slightly smaller 114 mm versions are better. But the PowerSeeker line is nearly entirely composed of low-quality equipment that is not worth purchasing.
But even before purchasing, joining a club is the best thing you can do. Some clubs even have loaner telescopes that members can borrow to learn before they buy.
I wish you luck with whatever you end up purchasing. Clear skies!