Okay, as predicted, here's the wall of text!
Okay, Bowser is very underestimated in this game. He's way better then he was in Melee, but a lot of people don't seem to notice. Bowser is a beast in this game, in the most literal sense.
Bowser's ground game is excellent. His Ftilt outranges every non-projectile ground attack except for Mario's Fsmash. The Ftilt comes out quick, has a decent amounts of priority, and it will probably be used for spacing. Ftilts can be angled upwards to try to intercept Mario's aerials, and it does a decent job at that, since Mario lacks any sort of disjointed hitbox in those attacks. Bowser's Jab is also used for spacing and is one of his best moves. Bowser's Jab comes out quick, has nearly no cool down lag, and has good range and priority. (It actually outranges Pit's Fsmash and clanks with it.) Jabs will most likely be used when the Ftilt would be too slow to punish Mario, or if Mario's close to Bowser. Bowser's ground game simply outclasses Mario's in the end. The good news is that you can pull off some pretty cool Cape--->Fsmash stuff against Bowser's attacks if you're ballsy enough.
The Fortress should also be mentioned as well. Bowser's Fortress, as most of you know, can be used OoS and is simply an outrageous move. It has invincibility frames at startup, and a crapload of priority once the hitbox comes out on frame 6. The Fortress also does a decent amount of damage and will probably kill Mario at 120% fresh, so you should watch out for that. Mario is a bit more susceptible to an OoS Fortress compared to some other characters because most of his attacks lack disjointed hitboxes, meaning he's more easily hit by an OoS Fortress. An OoS Fortress can be avoided by properly spaced Bairs, which you should be using anyway. The thing you really need to keep in mind is that being overly aggressive in this matchup is a bad thing. Bowser simply ***** overly aggressive players with the Fortress. The good news is that Mario really isn't an overly aggressive character anyway. Mario has good keep away game, and you should be exploiting that to it's full potential. Your Fireballs are awesome; use them.
Bowser's grab game also needs to be explained. Thankfully, Bowser's grab range sucks. It's simply bad. Really bad. However, if Bowser lands a grab on you, Mario's in a very very bad situation. I think everyone knows by now, but Bowser has the most **** grab releases in the game. Bowser's grab releases on Mario are pretty basic, but Mario can still get hit by a lot of stuff. If you ground release out of Bowser's grab, he can simply grab you again, pummeling you for more damage. If you're near a ledge, Bowser can use his Dtilt out of a grab, and that's a potential killing move. And if Bowser just wants to smack Mario for some beefy damage, he can use his Klaw out of a ground release as well, which will do a pretty hefty 18%. It's never good to get a ground release, since Bowser can always just grab you again once you break out. Getting a jump release is a better option for Mario, but Bowser can still hit Mario out of a Jump release with an Fair, which can kill at the ledge, or a Klaw, which will still do a chuck of damage. So all in all, it's a bad situation either way, but at least you're free from Bowser's chaingrab if you jump out. Basically, just avoid the grab as much as possible. It *****, but it shouldn't be that hard for Mario to avoid it. Just play keep away with your Fireballs. It's annoying and it works.
Bowser's aerial game is in most respects, pretty unimpressive. However, it's still something you need to look out for. Bowser is a character that does much better when he's underneath his opponent, and that's why he does so well on Norfair and Battlefield. If you're in the air and Bowser's below you, try to get to the ground as fast as possible. Otherwise Bowser could jump up and try to Uair you. Bowser has one of the most powerful Uairs in the game, and if you're in the air, it'll probably kill Mario at 80%. It will also go through all of Mario's aerials, so make sure you avoid this, since dying at 80% sucks. Bowser's Fair and Bair come out quick, and can be used to clash hits with Mario when you're both in the air. The Fair is pretty basic, but the Bair is pretty deadly. It hits hard, and it's slightly disjointed, giving it a pretty hefty amount of priority. It will probably be used as an edgeguard, so watch out for that, considering it comes out quick.
Oh, and you need to beware of Bowser's Utilt. I always dedicate a paragraph to the Utilt because it's so awesome, and it's probably one of Bowser's best moves. The Utilt comes out very fast, it has lots of priority, and it has an extremely bogus hitbox. I always say the hitbox is bogus because it feels like it's glitched sometimes. The hitbox is so large that it beats out all of Meta's aerials. That's how weird it is. Mario is no exception, and the Utilt will cut through every aerial in his arsenal except for maybe a well spaced Cape. The Utilt is even deadlier on stages with platforms, since Bowser can fire off Utilts much easier when platforms are around. Oh, and the Utilt will kill Mario at about 110% with good DI, so you need to watch out. It's an excellent killing move, and it's very hard to see it coming.
Of course, Mario has some good advantages in this fight. His keep away game has been mentioned, but as you know, he can rack up damage pretty quick. Utilt combos and aerial juggling simply destroy Bowser, since he doesn't do well in air and he has a blindspot below him. The blindspot exists because Bowser's Nair and Dair are so crappy. Just get him into the air, get underneath him, and begin the ****.
Mario can also gimp very well in this game. However, it's a bit harder to gimp Bowser then most people think. Because Bowser is so heavy, he's very difficult to get off the stage. And if he does get off the stage, most Bowser players will probably DI upwards to the corners. Bowser actually moves very fast in the air for some reason, and if he gets blasted to the corners, he can usually make it back to the stage without even using his Fortress. If Bowser recovers low, then just FLUDD and Cape him. Free gimp.
I still think it's a 50-50, just because Bowser dominates on the ground, but Mario dominates in the air and offstage.