Dr. Mario is in reality WAAAAY different from Mario. The pills are probably one of the major differences, mostly because of their knockback on contact with your opponent. But there are others.
Dr. Mario is actually slightly slower (almost unnoticible, but does affect his gameplay). He's got a stronger side-smash attack than Mario, but it's got shorter range. His up-Smash attack actually has the ability to bounce an opponent off the floor if caught in the correct frame, which can set up for combos. Mario doesn't have that ability with his up-smash. Dr. Mario's aerial attacks are also way stronger, most notably his back-kick. However, his punch does not cause a meteor attack, but Mario's does. Lets' also not forget about his down-tilt, which scoops the opponent up and flings them backwards and downwards, which is great for getting people off the edge. Mario's down tilt serves more as a means to lift an opponent into the air slightly to set up for a combo.
As for special moves, Dr. Mario's cape actually gives less lift when used in the air. It's not good for recovering to the edge, but his cyclone (down+ is very good (for those that don't know, when using either Mario, you can start the cyclone, and consistently tap B to lift into the air... Doc does it better).
Those are all the differences i can think of. It's mostly suttle things, but they all add up to create a totally different player. Doc is far from useless, and most tournament players consider him to be on a higher tier than Mario himself. The entire purpose of clone characters was to give similar, but different gameplay experience to vary up the gameplay. That's pretty much why you couldn't combine Fox and Falco, I would get into all the little differences there but i dont' really have time.
