I'm lucky enough to be seeing both this week before they close. But if I could only see one I'd choose Finian's Rainbow.
The reason I'd choose Finian's is the fact that I'll probably never be able to see a professional production of the show in the near future again. While I'm sure this Ragtime is extra special I have the original production fresh in my mind and regional productions of this show pop up frequently.
I really think you should spend whatever you have to and see both Finian's and Ragtime. They are some of the best revivals I have seen in years. If at all possible, skip In The Next Room. Really. If you have the chance to see these two glorious productions do NOT miss it!
If this is not possible, I would probably say choose Ragtime if you want to leave thinking about what you saw the rest of the day, possibly with a tear in your eye. See Finian's Rainbow if you want to leave the theater much happier than when you entered, with a grin on your face and a song in your heart.
Finians for sure. Ill stress what others have been saying: Ragtime has/will continue to have a healthy afterlife. As far as Finians is concerned, the show is produced so few times and is pretty unpopular, so when are we going to see another full scale revival any time soon? If I had to take a gander I would probably say never. So do your self a favor, go support a classic gem of a show that wont be around again for a long long time.