Like dapperlace said, both games are actually very different. Fire Emblem Fates is a Strategy RPG which is similar to playing chess. Bravely Second is a turn-based RPG which is similar tp Final Fantasy and Dragon Quest.
Bravely Second continues from Bravely Default iirc, so I don't recommend picking that up unless you've played and enjoyed Bravely Default. It's a great game if you enjoy traditional turn-based RPG with a twist. While I didn't like the plot progression in Bravely Default, I really enjoyed the gameplay and had fun testing out classes and how it works. I'm also a big fan of turn-based RPG so picking this game up wasn't a hard decision for me.
Fire Emblem Fates is a good entry point for new players, or if you've played any previous Fire Emblem game, would also enjoy the game as it's geared toward seasoned players as well. Unlike Bravely Second, FEF requires more planning and management. It does have Phoenix Mode so if strategic gameplay isn't your forte, you can still enjoy the game. FEF also offers a big cast and all the interactions between them.
So, should you get Bravely Second or Fire Emblem Fates? I say you should get
Fire Emblem Fates for now as you are interested in the different storylines, and the game was pretty enjoyable for me too.
I haven't played Bravely Second so I can't compare it with FEF yet, but I'd only recommend getting it if you have played Bravely Default.
Hope you'll have fun with whatever game you decide to pick up!
