There has been a recrudency of R.E.M.-U2 lately because of the stated 'invasion', but it wasn't the stqart of it all. I think you're quite right to say that the two bands are similar - their history, friendship, evolution, political involvement. They're both 80's college alternative bands that broke out in the mainstream. As far as 'alternative' music works (at least, worked past in the time where the word alternative meant something), they are two of the most accessible bands of this movement, with a long history, and respected in the musical world.
I think it's exactly because those two bands are similar, that they are compared very often, that it get on everydoby's nerves. There has been countless U2 or R.E.M. threads asking which is better. Not only is this a bit stupid (at least, here on Murmurs), but it's also divisive. There is probably also a bit of jealousy about U2 getting more success (especially in America) than R.E.M.
Yet, overall, most R.E.M. fans respect U2 (some like, some dislike, and a few hate - but it's the same with almost every other band), and this board is no exception. But there are some friction at times. Some like to get into these fights.