First, I am a born and raised jazz drummer. I don't try to pass that off as a badge of honor or anything, it is just a fact. I learned jazz first and then branched out into other forms of music from there. I got my first paycheck for a jazz gig at around 13 if I am remembering right. What jazz did for me was teach me about listening to other players. I learned how to play with touch and how to help other musicians build energy in their solos. Jazz also has a different feel to it than straight ahead 8th note based rock and roll. Think dotted quarter note, dotted 8th note and triplets for jazz. I can play straight ahead rock, but a lot of the jazz feel oozes into my playing whether I like it or not.
Those are the advantages. Here are a few problems that can develop from jazz training. I have to look at my playing over the years honestly if I am going to improve. These are some of the insights I have gained from long and hard critique of my own playing.
1) Sometimes I tend to overplay when playing straight ahead rock or blues tunes. This includes all sorts of things like playing 3/4 over 4/4 feeling fills etc. If I am playing with other musicians who are used to that feel it doesn't cause any problems, but if I am playing with straight ahead 8th note players it can really screw them up. If you grow up playing jazz you have to watch for this constantly.
2) I have jazz foot. I have to constantly pay attention to how hard I am hitting the bass drum. In jazz I use the bass drum mainly for accents here and there. I also hit it over a range of dynamics so one hit might not sound the same as the next. The feel I developed is great for jazz, not so great when the bass drum and snare are providing the main feel of the song.
3) I have jazz hands. I have the same problem with the snare drum as I do with the bass drum. I have to concentrate on hitting the snare nice and loud and at a consistent volume or the backbeat tends to get lost in the music. I also tend to use the ride cymbal or hi-hat as a main voice to drive the groove and it just doesn't work so well for rock/blues.
4) Because I am used to listening to other players rather than just laying down a groove, I can sometimes let the tempo vary a bit. Like it or not, if you listen to what the bass player and the guitarist are doing, you will tend to adjust your playing to them and the tempo can get away from you in a hurry. ALL bands do this when playing live so I am not sure what the best solution is. Even Steve Gadd does this with Clapton.
5) I haven't been elitist in the last 15 or more years, but really good jazz players can tend to look down on other drummers. In my opinion, that mentality is a mistake that will hold the jazz drummer back. It is best to keep an open mind. You can learn something from every drummer you go see.