Planets are those things that got a really crappy definition by the IAU a couple years back. One feature of this definition is that they have to be massive enough to collapse into a rough sphere (hydrostatic equilibrium - if it's rotating, it can be fat at the equator).
Comets have no official definition. Nor do asteroids. So both could be either. As it turns out, there are objects classified as asteroids that clearly had a tail at one time. Indeed, one of the annual meteor showers has particles that clearly come from an object that we currently call an asteroid. These particles are thought to come from the objects out-gassing, which would have formed a tail.
To make the definition matters worse, asteroids are officially cataloged as "minor planets". Two of the minor planets, 1 Ceres, and 134340 Pluto, also have the classification of "Dwarf Planets". Clearly, not all asteroids are Dwarf Planets, but at least some Dwarf Planets are asteroids. So one could argue (and i'm doing that now) that comets are closely related to planets.
However, comets are generally described as objects that do out gassing, sometimes have tails, and so on. They're not generally described as "spherical", and indeed, the ones whose cores have been imaged look something like lumpy potatoes. But planets can and do have tails. Planets with atmospheres have some bits of their atmosphere blown into space by the solar wind, etc. For some of the hot Jupiters found around other stars, this super heated atmosphere is blown out into space as a very visible tail. I don't expect an object with Jupiter+ mass to be called a comet, but i do expect the IAU to completely mess up the definition of comet when they get around to it.
It appears that finding a large object causes astronomers in general to become stupid. We got poorly defined minor planets soon after Ceres was discovered and studied for 50 years. We then got poorly defined planets when Eris was discovered. Stupidity like this can only come from large committees. Group intelligence seems to go down as you add more people. So even a group of PhD astronomers, when taken by the thousand, are incredibly stupid. So imagine Congress, where a PhD isn't required. IMO, the IAU should be disbanded, and replaced by a peer review system. Peer review, at the very least, is fairly fast, and has been shown to work. It also has self checking for those cases where it doesn't.