Okay, quick matchup summary coming!
Okay, first of all, you need to avoid Bowser's grab. This is a problem for some characters, but not for you. That's because you are Peach, and you should not be grabbed. If you're not floating in this matchup, then you're doing it wrong.
What sort of horrors can Bowser unleash out of his grab? Grab releases. If Bowser grabs Peach and you get a ground release, he can grab you again for more pummel damage and he can even use a Klaw, which will do a sizeable 18%. You fare a little better if you jump out of Bowser's grab with a jump release, but it's still not a very good thing. If you jump out, Bowser can Fair you, which can KO at the ledge. However, you should not be overly scared of this. Remember that you are Peach, and if you're using Peach, you are floating. If you're not floating, you're doing it wrong.
Bowser also has a very large shield, which he an use to absorb attacks and counterattack with his OoS options. Large shields are not a problem for you, because you are Peach. If you're not floating and Dairing/Fairing your opponent, you're doing it wrong.
Bowser's OoS options are pretty beastly though. His Fortress has invincibility frames and comes out instantly. If you're not floating at head height, which you should be, you'll probably be hit by it. Just make sure you're floating high enough for this to be ineffective. Then again, I think the hitbox might have some vertical disjoint, so I'll test it to see if the Fortress can stop Dairs and such. (probably not, but I mind as well)
Oh, and there's a move that deserves it's own paragraph. The Utilt. Bowser's Utilt is absolutely beastly, and it's one of the most powerful Utilts in the game. It comes out fast, has lots of priority (it will beat all your aerials), and a very bogus hitbox. Did I mention it'll probably kill at 110%? The reason I mention this move is that it's simply too good. It's very hard to predict, and it's one of Bowser's most usable killing moves. Watch out for this, especially if there are platforms around.
Other then that, watch out for the occasional aerial and Bowser's edgeguard shenanigans. Make sure you save your killing moves, otherwise he'll be a nightmare to kill.
So yeah, I would agree on something like 60-40. I might go as far to argue that it's 55-45, but that seems a bit low.