Doc's mobility is slower all around in all parameters, other than falling speed. Your recovery will always be poor (and the Doc's sheet no longer stalls in the air), but you can take advantage of that falling speed in terms of getting some vertical "reach" on your aerials and side specials. The Doc also jumps lower than Mario which is also not too good.
Mario is a better gimper, but the Doc's tools are better for on-stage killing. If you can, a well placed bair can ruin an opponent's recovery, but remember that you have the power that Mario somewhat lacks.
D-throw's comboability is reduced from Mario's version, but his other throws are more powerful and can function as launchers.
I've found that the Doc's F-Smash does more damage the closer the Doc is to his opponent. While both parts can kill, if you can get into an opponent's face, that's cool.
Mario is a combo god. The Doc is not, though he can manage simple combos. The Doc's forte lies in his greater power so you should exploit that to get earlier kills safely from on-stage.
Unlike Mario who will use his fireball for approaching, the Doc's pills are best used for stage control. If you need to catch some breathing room, use pills. Rather than approaching the opponent with your poor mobility, it might be best to let the opponent come to you the way you want to, and pills help with that.
To help Dr. Mario's lousy recovery, use the Dr. Tornado as you're jumping and mash the button to get more height. The move can also be used for occasional attacks and works alright to rack up damage.
The Doc's Super Jump Punch is now a single-hit move with a sweet-spot at the start of it, not a multi-hit move. It complements the Doc's strength, so if you can hit with the sweet-spot, that'd be swell.
As said above, Doc's sheet no longer stalls in the air. While this nerfs his recovery, it does give you some vertical advantage in terms of reflecting projectiles back and flipping recovering opponents around.
The Doc's dair has a spike sweet-spot, I believe. If you can reliably hit with it, it could be used to get some early kills, though remember that Doc has a lousy recovery and that Bair and Sheet will be your main gimping tools.
That's all. I'm not one of the people who main the Doc, but I hope I helped you out all the same.