It is extremely important. I personally hate buying Nvidia cards now because I feel they starve their cards in terms of memory bandwidth to help keep performance gaps between their own cards. AMD seems to have the bandwidth much more reasonably placed on their cards to me.
If you want a good example how important it is though, look at comparisons of the 7790 and the 7850. The 7790 is probably AMD's most bandwidth starved card. It is very close in shader count to the 7850 so the cores should perform very close, but the 7850 has double the bandwidth and so it performs a lot better.