You know what bugs me? There is SO much romance in this fandom, but hardly anyone explores the other relationships. And for someone who's as big a fan of friendship and bromance as I am, that sucks. (That being said, this isn't slash.) There's an ok amount of Beast Boy and Cyborg friend-love out there, and plenty of Raven and Starfire, but other than that...neeeh.
I find these two interesting. There is so little anywhere about them, and yet you would THINK that they would get along so well considering how alike they are. (Yes, I said ALIKE).
Think about it. They both watched their parents die, and feel responsible for it in a way. They both have a somewhat difficult adopted father (Batman, Mento). There is actually so much of their personality that is similar, though it's only obvious when Beast Boy is being serious.
But of course, they don't know all that. And all they're really aware of are their differences.
I think that they actually do care about each other, but one thing that keeps them from being very close is their way of showing it. In the show you do see little things that Robin does for BB, and that he does respect him (It was for Beast Boy that Robin attempted to give Terra a second chance.) But he's very... well, he was raised by Batman.
Beast Boy on the other hand seems more like a touchy-feely kind of person. He has no problem showing his emotions (if they're positive) and letting people know how he feels about things. He also seems to be a lot more physical (not in a weird way, but in a huggy kind of way. Just like a lot of people with outgoing personalities.) He's always hugging and hanging on people.
I think Beast Boy takes Robin's standoffish attitude as him not caring, or being condescending. And Robin takes Beast Boy's bubbly-ness as an intentional annoyance.
I do wish their friendship could be explored more, though, because I don't think it HAS to be like that.
Robin and Beast Boy bromance. You're welcome.