I've heard that the only difference between a star and a planet is mass, meaning that if planets accredited enough mass they would too become stars.
Regarding the title: Yes.
Does this mean that the star started off as a planet?
Yes, a star could technically start out as a planet, if it accreted enough mass. However, this is extremely unlikely, since the planet would need to be 80x the mass of Jupiter for it to undergo nucleosynthesis.
Stars require hydrogen fusion and earth has little H. Could the earth become a star if more mass was added, but its relatively low abundance of hydrogen remained the same?
The most important components for a star is the composition and mass. As it stands, Earth doesn't have nearly the concentration of hydrogen for a star to form, even if we kept adding mass.
Most stars work by fusing the lightest element, hydrogen. To fuse heavier elements requires a much, much higher temperature, since the more protons the element has, the more it will repel other ones instead of fusing. Yes, massive enough stars can fuse heavier elements.
However, Earth would inevitably grow in hydrogen, so it would eventually have a significant amount. Above a certain size, given ambient conditions (like radiation pressure, light intensity from the star, etc), Earth would become massive enough that it starts capturing hydrogen gas from the interplanetary medium, and becomes a gas giant (as called2voyage pointed out). Where this happens isn't precisely known, but it's roughly 10x Earth's mass. This means that, theoretically, a terrestrial planet could form a star if you increased its mass ever further.
Now, I can hear you thinking, "What if Earth were surrounded by a vacuum instead? If we kept adding mass, would it become a star?" Well theoretically, the planet would eventually reach temperatures where most of its elements could fuse. For example, the "star" could sustain itself on oxygen and silicon fusion, but not on iron fusion.
A star does not start off as a planet; you have a large cloud of gas that is collapsing in on itself due to gravity. The majority of the gas goes towards creating the star (more than 99% in the case of our Solar System). However, gravitational collapses can occur several places in the gas cloud, and some of the gas will contribute towards the collapse of far smaller over-density seedlings. If the mass of the smaller collapsing object is large enough, gravity will pull the object together into a sphere, satisfying one of the three criteria we have to classify an object as a planet (see e.g. this question for more information about such criteria).
The very large mass of the gas which is going to become a star allows it to collapse in a way where the pressure becomes great enough that the central parts of the cloud start fusing together hydrogen -- or burning, if you wish. You do need simple gases for this type of ignition to occur, as even higher temperatures are needed for more complex atoms to fuse together. This is because fusion becomes less and less energy effective as the atoms become more complex.
While I have no direct sources to refer you to, you can be decently sure that the Earth wouldn't become a star if it increased in mass without increasing it's amount of hydrogen substantially. However, with the great amount of hydrogen present in the Universe compared to the other elements, a scenario like that would not occur, and Earth's hypothetical accumulation of matter would realistically be of gas which would be used for the effective fusion we find in stars.
No. In order for a star to sustain itself, fusion must take place to avoid collapse due to gravity. The earth is made from heavy elements (nickel, iron, etc) which are nearly impossible to fuse in stars. Therefore, due to this, the Earth cannot be a star due to the addition of more mass.
But, if the additional mass was hydrogen, I'm conflicted to say what will happen, perhaps it'll be a low mass star that will shine for a limited time until it cannot sustain itself or fusion just might never happen.
