There is only one band I can say are similar to U2, though many may not have heard of them. They are a Christian band called Delirious. I often see them as the Christian version of U2 (even though 3/4 of U2 are Christians anyway, I think). They're singer has a unique voice that doesn't exactly sound like Bono's, but sort of touches you in the same way. It's hard to describe. Their latest album is definately worth checking out, if you're of the Christian persuasion (or even if not, many of their songs can be taken as well-crafted pop songs and nothing more).
I wasn't around when Boy came out, but I can't think of any of the music of the time that was like it. I only purchased it recently and love it, it has a punky edge to it without being punk, and is very dark in tone without being depressing. I think the coupling of Bono's lyrics and vocals with Edge's guitar (not to to disservice to the other members, though) is just a totally unique blend that is nothing like anything else in the music world.