If you look at the set up of a rim the is a small box under the extension off the back board. It serves as a pivot point on the rim. Inside the box there are a few springs that can be either adjusted(either looser or tighter). Any way I think alot of the idea of a soft rim is that the springs are looser than average thus giving the ball a better chance to rattle around and in. If a rim is very hard you will see alot of long rebounds espiecally if the one of the teams is really tired(this usely means they aren't getting the lift in there jump shot and shorting the ball). Also you will see a slightly lower scoring game on hard rims(maybe 2-6 for each team, if that). The opposite will happen for a soft rim(marginally higher scores, shorter rebounds, etc)
Overall it won't make much difference is a game. The bigger difference is if you have a solid back round to shot against. If you don't believe me set of a clear backboard outside sometime and try shotting, I promise you, you will miss the first few shots finding your range. Now will this have a huge affect on the BTT, no, but is would be a huge deal for the teams in the Final Four. The Final Four is being held in Ford Field using the whole stadium for seating a backround will be insanely important then, very important if a team didn't take enough time getting use to shooting against nothing. All in all shotting comes down to mechanics much more than the rim.