Just to start off, Dr. Mario's moves and attributes make him noticeably slower than Mario. As a rule of thumb, his moves are stronger, and I think each of them come out at the same time. If all of the moves themselves are the same speed, aside from his D-Air and D-Special, then you should respect them more than Mario's.
Because of how he handles, and while this doesn't necessarily relate completely to gameplay, I think most players will prefer to stay on the defensive, or play more cautiously when approaching. Note: Whenever I say something that isn't necessarily related to gameplay, it'll be in asterisks, so here's an example:
* This character benefits much more than the plumber from being able to read and acknowledge whatever the enemy is doing *
First off, when playing against him, don't treat him exactly like Mario. Yes, they look very similar, in moveset and in name, but they perform much differently. With his moves being more powerful, you shouldn't treat him like a joke. You wouldn't do that to Jigglypuff, would you? (If it was a bad player then yes, but moving on)
• As stated before, slower, and generally more powerful than Mario. Jumps are lower, but in turn this gives him better short hops and near-ground aerials than Mario.
Grounded Attacks
• U-Tilt launches you perfectly upwards, allowing it to still combo into itself at low percents, and still lead into U-Air at mid percents. At around 80%, he'd have to double jump to hit you with an U-Air. Just be wary of his grabs so that you don't get suckered into this; None of his throws set it up quite enough to the point you can't just strafe away from his U-Tilt, but it does put you in a bad situation, right above him.
• F-Smash has a tad less range, and the sweetspot is actually near Doc's hand, instead of the blast itself. As always, be careful with this attack. A pivoted one might be able to punish your rolls, since R.O.B does have a fairly large hitbox.
• D-Smash, while just a bit more powerful than Mario's, can kill near the ledge even with no charge at 130%. Even if it doesn't, it can still put you into a bad situation where you're getting edgeguarded. Up close, I think this is one of his stronger options, so try and wall him out with F-Tilts and D-Tilts, since neither of his can outrange yours.
• U-Smash has a totally different trajectory than Mario's, launching you diagonally. Interestingly, it launches the opponent in the opposite direction of Doc - A backwards hit makes them go forward, a forward hit, vice versa. While it's very powerful, and could send you toward the ledge if you get caught by it, and has good range behind Doc, it requires you to know his differences from Mario to get the most out of it, and is limited in use because of it's two hitboxes launching different ways. Either way, try to avoid rolling near the edge, as you might get hit by the backwards hitbox.
• Dash Attack pops you up into the air at 0%, where he can follow up with N-Air and U-Air, instead of Mario's diagonal trajectory. Watch out for this move. It instantly sets you up a fair amount in the air, like Samus's does, where R.O.B definitely should not be. At this point, he could follow you with a D-Air, an Up-Air, or Dr. Tornado - Super Jump Punch if he's literally right next to you. It helps his approach much more than Mario's in the early game, whose dash attack just sets them in a position where they cannot be attacked, and makes you unable to use your projectiles effectively. Punishable on shield.
At around 80%, this move requires that you double jump to immediately hit R.O.B with an U-Air, so it becomes less useful than Mario's as damage starts to pile up.
Aerials:
• D-Air is different, similar to Luigi's and Falco's, where they spin with their feet pointed diagonally. Also has a sweetspot at the tips of his feet. Doc can use this with a short hop to approach you diagonally and very quickly, although it's awkward to do this without leaving yourself open to an U-Smash or U-Tilt from R.O.B. An angled F-Tilt should also do the job against him. In the air, he can also use this very handily for quick damage, and to beat out your N-Air. I think he could potentially mix this up with his N-Air while trying to land, if he doesn't use Dr. Tornado. With R.O.B having such a large hitbox, try to avoid this, as it's free damage for them.
• N-Air does more damage the longer it's held out, encouraging people to use it while falling. Try not to get caught by a fastfalling one, as it can do 9%, not bad at all for an attack with so little range. Could potentially beat out a N-Air and U-Air if they're not spaced properly. Also good for breaking out of combos, but since R.O.B doesn't really have combos besides his U-Air and situational F-Air, it'll mostly be used as chip damage and a quick "Get off me" move.
• U-Air's trajectory, instead of popping the enemy upwards, sends you left or right diagonally depending on whether Doc hit you from the front or behind, although it seems to have a slight chance to hit in the opposite direction if you're very close. While it doesn't have much potential to combo into itself, it can lead into his back aerial, or potentially used to edgeguard with the back hitbox. It's not very useful past 80% or so for following up moves, limiting how effective it is.
• F-Air, the punch, is much different than Mario's, not doing a set amount of damage with early/late hitboxes and a sweetspot, that being Mario's fist in front of him. While the sourspots only do around ten or so percent, the sweetspot launches enemies diagonally and upwards, regardless of whether they were hit on the ground or not, with no spike downwards.
Testing on R.O.B, from the middle of FD, it killed him at 120%, down to 100% near the edge. With this move being more versatile, it's basically like Ganondorf's F-Air: It could make for a very surprising kill if you aren't prepared.
• B-Air is the same as Mario's, with one crucial difference. It actually kills at reasonable percents, dealing more damage and knockback. It seems to kill R.O.B in the air at 120% or so percent, although it puts you in an edgeguarding position much earlier than that. While it's fairly predictable as a kill move, it comes out extremely quickly and has very little endlag, making for a mix-up between his F-Air and B-Air for kill moves. By itself, too, 14% isn't bad at all, so watch out for it even when you're at low percents.
Specials:
• Neutral-B, or the pills, is similar to Mario's, but deal more damage, bounce only two times, and have a different trajectory, bouncing in a more curvy motion. To be, though, the major difference is that they go a much shorter distance in the air than Mario's, which is a good thing for him against R.O.B. Dr. Mario could potentially stay just out of range and pepper you with pills, dealing chip damage that slowly gives him the advantage in the beginning of the match. He definitely has a projectile to keep us honest.
• Side-B, Super Shock, doesn't stall Doc in the air, has less horizontal reach with more vertical reach, and does slightly more damage. Overall, it makes him just slightly safer than Mario on the ground, as angled lasers can sometimes be reflected by the upward hitbox, and helps protect against gyros as well.
However, since it doesn't stall Doc in the air, coupled with his slower air speed/lower jumps, his recovery is just not as good as Mario's. If you can get Doc out a fair distance horizontally, he's toast. With the extra vertical reach of the cape, though, a Doc can actually settle down at the ledge and potentially reflect your gyros back, saving him some trouble.
• Up-B, the Super Jump Punch, has a fair difference from Mario's; It's a single hit, like Luigi's, but not impractical to land. Doing around 14% or so percent, it's a great OOS option for Doc at late percents, knocking opponents back, and kills R.O.B at around 130%. For a move that has a good amount of range, can actually be used out of shield (with tap jump on, I believe) and serves as another kill option for Doc while still being a recovery... It's extremely good.
Also, once the initial hit connects, there's a small window where Doc can input in the initial direction to change where he's facing. So, while it's a very silly idea, he could potentially Up-B you off-stage and still recover.
• Down-B is like Luigi's, except it travels a much shorter difference. This is often used for Doc's recovery, since he can repeatedly press B to increase it's distance in any direction, with a jump still left if he hasn't used it, and Super Jump Punch. If the Doc knows what he's doing, he can potentially make it back, although I'm sure Dr. Tornado is punishable with a D-Air.
Note: It actually does destroy a fully charged gyro, I think, although it decreases the distance it travels, if my test was right.
So, be careful and don't waste your gyro when he's using it.
* If he's using it at all on the ground, you probably won't have time to use a projectile anyways *
Similarities:
Well, you all probably saw them before anyways.
With all of this being said, a Dr. Mario has plenty of options to rack up damage, keep his enemies out, and the luxury to play a little more defensively with his Super Shock and Pills. Lasers should be as effective on him as they are on Mario, same with gyros, as they won't always have the opportunity to reflect, but since he has a decent time in the air, he can likely jump to avoid them. I think our N-Air is slightly less effective in this match-up, because he has five tools to defeat it, (D-Air, Dr. Tornado, N-Air, and potentially fastfall F-Air, not to mention pills) and can simply rain Pills on us from above.
This being said, we do generally outrange him on the ground, with our tilts going further, and being able to play defensively ourselves with Gyro and Laser. Doc isn't excruciatingly slow, to the point where he isn't able to approach at all, but it's still enough to limit his options. You definitely need to respect all of his attacks, as they stack up damage on you much more quickly. Near the edge, he has a multitude of options to kill you with, besides the obvious smash attacks; While this is true for R.O.B as well, many of them are obvious, (His U-Air due to it's usage, D-Air since you'll be using it off-stage, and U-Smash) while Doc is fast enough to use all of them to his potential.
* If someone picks Dr. Mario, anyways, it's likely they have a more defensive playstyle, and consequently thinking more. If this seems to be the case when you're fighting one, you'll also have more time to observe how they're playing, and make adjustments to your playstyle/punish them as needed.
Summary: Treat Dr. Mario with extreme respect, especially his dash attack, as it leads to a load of percent on R.O.B if he can get aerials onto you. His D-Air is a quicker and less predictable option to approach, as it can be short hopped easily and used off-stage to edgeguard.
Watch out for B-Air, and Super Jump Punch out of shield, and remember that F-Air works differently than Mario's. The latter two moves can kill at reasonable percents for how versatile they are, while the F-Air can kill at silly low percents near the edge. That isn't even considering if the Doc has rage, in which case, they become more deadly.
Since his moves generally have more knockback, he can use them to try and keep you out. Unlike Mario, Doc isn't too wonderful up close, where he can be outperformed. Safe opportunities for damage aside, like when you're in the air, or caught charging your gyro, I think getting in with damage, and getting out with less yourself, is the mentality of a Doctor Mario player.
Dr. Mario is the most versatile clone in this game, and it shows. You have to have a different mindset when fighting him than with Mario, due to most of his moves having different properties, even if they look the same. Play smart, capitalize on his slow speed, and you can win.
(No rating, as I don't have much experience with this match-up myself to trust myself giving one... But I would either place it at even, or slightly in Doc's favor.)