In my own beers, I've gotten a little bit of flavor from the bittering addition. It's not nearly as much as a later addition, but it's there. I've done lighter beers with only a 60 minute bittering addition, and some of that hop flavor does come through in the finished beer.
Since you do get a little flavor from the bittering hops, I do believe that it matters what hop you choose as a bittering hop. The flavor is part of it, but there are also hops that are more or less harsh in their bittering. Something like Magnum, Horizon or Warrior are quite smooth. Columbus is pretty smooth too, but I think I get a little more "roughness" to it. Something like Chinook is even more rough. I've started using two bittering hops (e.g. 75% Magnum and 25% Chinook) to give a little depth to the bitterness. I like what I've gotten out of it so far.