This sounds similar to a storygame I'm writing where the character can be a human or an alien or a different species altogether. So yes, it's certainly possible, as long as you can handle all the changes and variables and stuff which arise.
As for effects, that would certainly depend on what your storygame would focus on. Special abilities is one that's been covered here, but then you have to decide when and where those abilities might come into play. If they're only in battle, then it might be easier. If they appear in special scripted events throughout the storygame, then...well, you may have to be prepared to write specific situations which use the abilities. It can get complicated, but still quite possible!
I would also give the player's chosen race their individual strengths and weaknesses. Perhaps humans have more luck and are more rounded as a character, whereas the aliens are faster and have the benefit of psychic powwaaaas.
Extra idea of mine: Maybe the aliens can fly from place to place, or possibly teleport, thus making travel easier. (That's assuming transport is relevant to the gameplay). To do so may exert a price on their stats, however.
Maybe I'm just looking too much into this.