This questions can be split in two; for planets and satellites.
The diversity of planets reflects in part the diversity in terms of chemical composition of the protoplanetary disk. We know that UV radiation from the sun can dissociate complex molecules or even very simple ones; for example, when UV rays split water molecules the result is free hydrogen and oxygen atoms. Since hydrogen is extremely lightweight they can be transported easily on the flux of stellar winds. So water, to keep with that example, if close to the sun could end up been dissociated and depleted from the region of the disk, but above the so called "snow line" UV radiation from the Sun was so weak that this couldn't happen as often and thus water molecules (which are very heavy in comparison to single hydrogen atoms), stayed there. That only explains the dichotomy between interior and exterior planets in terms of water content, and even then, some processes (like the late heavy bombardment) might add some water to the interior (as it happened on Earth). But this reasoning is not only for water, carbon dioxide, ammonia, methane and hundredths of different molecules have their own "frost lines". Closer to the sun the carbon can't be methane is a volatile gas that gets quickly pushed outwards, but at some tenths of AUs methane can remain in stable conditions and can even condensate into liquid droplets.
All of this just to say that the protoplanetary disk was NOT homogenous in terms of chemical composition, and was not homogeneous in terms of density or pressure. The thermal and chemical gradient across the nebula ensures some diversity and complexity for the entire planetary system.
Here you have a beautiful diagram showing how different chemical compounds could condensate in different temperatures and pressures on the protoplanetary disk.
Also accretion of planetesimals is more energetic closer to the Sun (meaning that breakups can occur more frequently and is difficult for a planet to grow large), while in the outer regions planets can increase in mass with regularity since collisions with other planetesimals are performed at lower relative speeds (because of how two similar orbits have a difference in periods that gets larger when you get closer to the Sun and thus larger relative velocities). This coupled with the gravitational interactions of the protoplanets and the early disk (see planetary migration and nice model etc...) allow for different accretion rates and the accretion of materials of different composition of what was found in the original place of formation of a particular planetesimal. This also helps to keep a wide variety on planetary masses.
A wide variety in planetary masses is the starting point for a larger variation as the planets evolve in time and diverge from their initial conditions. A rocky small planet (Mercury) might have less heat trapped inside than a larger one (Earth), due to the smaller energy released by smaller accretion rates. Thus it might get cold rapidly and a magnetosphere due to a melted interior can't happen. The absence of a magnetosphere allows for solar wind charged particles to erode your atmosphere by sputtering. Instead on a planet like Earth, the larger mass has led to a melted interior that in turn generated a magnetosphere that lasted for billions of years, in Mars it lasted some time but now is almost gone so the atmosphere has also been almost destroyed. On Earth the presence of an atmosphere leds to all sorts of chemical erosions, and phenomena. Also, it's melted interior coupled with the specifics of its chemical composition and thickness of the crust allow for a mechanism called plate tectonics. Tectonics can't happen on Venus because the crust is not as thick (due to different composition) and thus it doesn't break apart in plates but just deforms and folds in a complex behaviour that is unique to Venus.
Also collisions with planetesimals can alter the future evolution of similar planets. Venus was probably very similar to Earth (similar mass, very similar composition and not so dissimilar temperatures as one might think) but their paths diverged completely as tectonics on Earth recycled the lithosphere and on Venus the carbon dioxide got more trapped in a greenhouse effect, and because Earth had a collision with another planet that have us our Moon, which is a mechanical stabilizer while a random collision with Venus (with different impact parameters) led to an extremely slow rotation and long days (but no moons). Longer days mean different insulation, and that changes drastically the climate of a planet. On Mars days are similar to Earth's but since it is smaller and the atmosphere has gone many things are very dissimilar from Earth. Also, Mars has no tectonics (the crust is thicker than Earth's and moves like a monolithic object) so there are few volcanoes and they grow huge (while on earth a single hot-spot generated by a mantle plume drills several holes on the crust as the plate moves on what is called a volcano chain) and there are stress faults like Valles Marineris (unique in the Solar System) that on Earth would have been relaxed by tectonic movements.
To see how different the evolution of two planetary objects can be just by making them different mass take a look at our Moon. It has the same chemical composition (it is a chunk from Earth in fact), it is basically at the same distance to the Sun as Earth, it lives in the same interplanetary environment (same solar radiation, solar wind, impact rates etc...), and still it is completely different. This is all due to mass! The moon can't retain a large atmosphere as Earth because it has less gravitational pull. The same temperature for our atmosphere there means that the particles reach escape velocity easily and start to escape from the gravity well. With no atmosphere, not internal heat, the moon lacks almost any type of erosion in across billions of years of evolution. Processes of erosion on Earth have made the diversity of geological formations to explode in comparison to those found on the Moon. Even then the moon has its own peculiarities and dynamical features unique to it.
Now we are getting closer to the satellites question. They in fact should look almost the same, since they are formed from very very similar material in extremely similar conditions. And indeed we believe that moons originally were very similar (for example the 4 Galilean moons). But Io is to close to Jupiter and the other moons interact with it in such a way that the geological processes are completely different. Water and volatiles evaporated quickly as it heated up by tidal forces from Jupiter. These tidal forces were not as strong in Europa since it is farther away, thus it only melted part of the icy crust creating an ice analog of plate tectonics which generated a plethora of diverse formations. Satellites evolve. Enceladus shoots jets because of tidal interactions and orbital resonances with other moons. Some moons like Japeto have a dual colored surface because of the material sprayed by Enceladus landing on on one of its sides. Some moons like Triton have nothing to do with the other because they formed in another region of the solar System and later got trapped by the gravitational pull of a planet (Neptune in this case).
As I mentioned before. Atmospheres (density, composition and pressure) are largely dependent on the mass of the planet or moon. Look at this graph:
It shows the velocity of gas molecules in relation to temperature of the gas. For larger temperatures gas molecules move faster. In a planet with low mass, the escape velocity is lower than one with a larger mass. Thus a planet closer to the sun (at higher temperature) needs to be larger in size if it wants to preserve the same gas molecules in its atmosphere as a planet that is farther away (colder). You can see why Earth's atmosphere could trap and retain water, oxygen, carbon dioxide, ammonia, methane nitrogen and other gases while it is unable to trap hydrogen and helium (because they are lighter and thus for the same temperature they can move as fast as needed to escape Earth). Meanwhile, the Moon, which has the same heat coming from the Sun as Earth, since it is less massive it can't retain almost any gases (maybe a bit of Xenon). Titan, is a huge moon thus it can retain many gaseous molecules like Nitrogen and Oxygen (those in turn make the pressure high enough to retain also volatiles like methane in a liquid form on the surface). But why doesn't Ganymede has the same atmosphere as Titan if they are basically the same size? Because Ganymede is closer to the Sun, larger temperature means that the molecules move faster and thus they escape its attraction easily.
As you can see the complex processes of atmospheres of a moon or a planet change everything (erosion, recycling processes, chemical corrosion, etc...) and in turn that diversity in atmospheres comes from a diversity of masses and distances to the Sun.
I think the Solar System is a chaotic system, dynamically, geologically, chemically etc... Chaos means that for an small difference in initial conditions the system will evolve in exponentially diverging different states. Planets and moons might have started as similar objects but history and the chaotic dynamics of the system have evolved into completely different environments. Not only that, but the truth is that planets didn't started as equals but were very different from the beginning, so imagine how far away is Venus to become a Titan, or an Io to become an Earth.
Also there are processes and conditions that are specially well suited for divergence. For example: Earth is very dynamic while Mars, Venus, Mercury, the Moon and others are totally not. Why? because on Earth water can exist in 3 different states of matter. We can find liquid water, water vapor and ice in different regions and seasons. And that is because Earth is at an average temperature and it's atmosphere has just the right pressure to allow this. Earth's conditions are very close to the triple point of water (where all three states of matter coexist), that's why we have a water cycle on Earth, with rivers and glaciers eroding the landscape and clouds regulating the climate.
Mars, Venus, Mercury, all have temperatures and pressures were this can't happen, not only on water but on many compounds present there. You know where this can happen? On Pluto! This was very surprising, Pluto shows a variety of terrains and geological features that exceeds all expectations. Now we know this is because Pluto is extremely dynamic (as Earth) and a lot of erosion and geo-chemical processes can occur, but this is not because of water (since Pluto has low pressure and low temperatures), but because of Nitrogen and Neon! Both elements have their triple point inside to Pluto's range of conditions and thus Neon rivers, Nitrogen glaciers and hazes are expected on this dwarf planet.
It is indeed an interesting question. How incredible are the laws of nature that allow for extreme variety even between brothers. I wonder how a planet around any other star might be, our simplistic categories of hot Jupiters, mini-Neptunes, super-Terras, etc... are just so primitive and restrictive. What wonders await us in this complex and diverse cosmos is beyond our comprehension.