In the interest of full disclosure, I am a UF law student, but I'd go to UF if I were in your position. (And I was, last year, when I chose UF over UM.)
The reasons are many:
1) It's still the best school in Florida. While the competition has tightened up in recent years, with UF law cutting its enrollment in the next few years that'll change a bit.
Don't underestimate UF's weight in all parts of the state, including the home territories of UM and FSU.
With that being said, it is true that you will lose some networking opportunities while in school. That leads me to number 2:
2) Cost. This is undervalued by many law students. You probably would pay double the cost to go to UM. Many people who go to a school like UF or UM and are 100k in debt are under more pressure to get into any firm, let alone a firm that they might like. While there's still pressure for someone who owes 50k to get a job, it's not nearly as stressful....
3) Do you really think all they do in Gainesville is play and talk football?
Gainesville has a lot of cool things to do if you're willing to look for it, from outdoors activities to partying. Also, the "big" cities are only 1.5 to 2 hours away, and I know MANY people who take trips throughout the year. I'm not originally from FL, and honestly Gainesville is probably one of the coolest places I've seen, if you keep your mind open. But to be fair, it's not for everyone, and that's cool.
Also, keep in mind that you won't be spending most of the time "living it up" in the city. You will be studying!
4) I'm assuming your scholarship is 5k/yr., contingent on maintaining a certain GPA or class rank?
UM did this for me, and while it wasn't a hard requirement, it helped make my decision. (If I lost the scholarship, I went from paying about $10,000/yr. in tuition for the first year to paying $35,000/yr. for the last two years.)
5) You can always network over the summer. Many students do that.
With all that being said, UM is a solid school, and choosing a law school is a very personal choice. If you have any more questions about UF or Gainesville, feel free to ask.