They are mechanically very different even though the premise sounds the same. To summarize what I'll type below, in essence DBD is a hide and seek game. F13 is a scavenger hunt under pressure.
DBD are quick games in relatively small maps, with a single objective of start the generators. Easy to pick up, good amount of progression and stuff, good amount of variety but it's not an overly deep game IMO (can be positive or negative to each person!). You also don't need mics most of the time. It's fun, but is pretty much a game of hide and seek.
F13 has a lot more things to do. More items, more objectives (call tommy, call the police, repair the boat or repair one of two cars being the big ones) and the maps are much larger. F13 is more of a scavenger hunt type of game, and has some hide and seek elements, but not as strong as DBD. It's lacking in enemy/map variety compared to DBD as well, but I imagine with DLC in the future, that's going to change just as it did for DBD. The other big aspect is teamwork is pretty much required with an emphasis on voice chat.
I'm a fan of DBD. Owned it for a while. In my opinion, DBD is the better game right now. F13 has the most potential, but needs more polish and content before it can take that top spot in my mind. In a lot of ways, F13 is where DBD was when it launched a year ago so I'm curious how it will improve over the next few months.