The museum that is located in the Old Capital building is superb. It has well organized and interesting exhibits about the history of Florida and its government. Although I was very impressed, my kids did not share my enthusiasm. But how many kids are impressed with museums?
It is definitely worth a stop if you are in Tallahassee. If you like it as much as I did, you can explore it for a few hours. If you don't find it to be interesting, you can be in and out in a few minutes. Admission is free, so you're not risking any money.
If the new (built in the 1970s) Capital building is open, cross the courtyard and go to the 22nd floor. The view is quite nice.