What exactly does the number of connections do for me? Do I generate more coins with more connections, or is it equivalent no matter what?

It seems that from time to time, Tor gets tied up and can go for a few hours with 3 or 4 connections. Then, when I restart bitcoin or restart Tor to refresh my connections, I can get 7,8, or even more (9 currently).
I haven't really been able to tell if coins generate slower or not with less connections, but I was curious if it does correlate. If not, how exactly will having more connections help me out compared to less?