To add to what people are saying here, Mainz is definitely tankier than Bayard, but it needs to be because it's less maneuverable. With Steering Gears mods, Bayard can be very wiggly and dodge some incoming fire that Mainz generally needs to tank.
Generally, I would say that Mainz has better armament. The ballistics are very good on her guns (but they aren't bad on Bayard, either), though of course Bayard is capable of a better fire rate. For the exact DPS/fire-starting potential, check out LWM's Mainz review since it has stats for both.
I play mostly co-op so one of the big differences for me is the torpedoes. Bayard's torps do a little more damage and have amazing firing angles and better range, but I really like that Mainz has two launchers per side.
Overall, I really like both ships, but despite being mainly a MN player I prefer the Mainz. Honestly, I like the Prinz Eugen more than either of them, but I think I'm probably in the minority on that one. I think they're both good, but they feel very different from one another.