Do you take your existing character and gear into the new map or do you start over?
You use an existing character, yes.
However, if you want to 'start fresh', so to speak, but still generally meet the recommended level for The Following(level 16), you can start a new campaign of the main game at pre-determined points in the story, where you'll be thrown in at the appropriate level with the ability to spec your character out how you like with the points it gives you. Obviously, the farther you pick in the game, the more points you'll have to spend. So you can basically just pick a point, start it up, back out and then use that character for The Following as well. This is useful if you feel you've got something of an overpowered character or if you simply want to spec your guy out a bit differently and not have tons of items and weapons already.
I will say this, though - the game is generally harder now(across both main game and The Following). Zombies move faster and are more aggressive. Sit battling a zombie for more than a few moments and you run a very high risk that more are quickly coming up on you from the side or behind you where you cant see. It also feels like they take more punishment than before, which will make it a lot more difficult before you get finishing moves. Running away just seems like the better option a lot more frequently now.
This may be annoying for some, but I think it's pretty cool. Along with playing on Hard mode and without the minimap on, I feel this may very well be the best 'survival' horror game yet(dont mean that in the traditional definition of the genre, but in literal terms). Not knowing where virals are ahead of time or when chasing you makes for some super thrilling chases, especially since it seems like there's a lot more of them in The Following. I came upon a group of five of them at one point. And yea, this applies to Volatiles too, which.........um......yea, if you're out at night, wear diapers. And dont think being in a car makes you the least bit safe from them.