My best friend of about 10 years is an Aspie and I'm now beginning to suspect that I might be too. (I only heard of its existence last month.)
I can tell you that I find it a lot easier to be around her than most anyone else. She and I share a lot of common interests. The most interesting thing about the friendship I've noticed, though, is that we can go weeks or even months without speaking to one another. But when we see each other again, it's like we were never apart and we haven't missed a beat.
When we're together, we talk about our interests, funny things we've noticed about life, and whatever else suits our interests. We rarely if ever talk about anything emotional or anything too serious. We don't talk of relationships. We don't talk about our problems. We never gossip. These things just never come up as a topic of conversation. There's no pretense. We mean what we say and there's no reason to fake who we are. (Mind you, I can only speak for myself, but I believe the feeling to be mutual.) We're also both a little socially clueless, but since we're both on about the same level, it doesn't matter.
We also coexist very well. For example, she might be on her computer and I might be on mine...there might be long bouts of silence, but it's never awkward. We also have no problem going out and doing things together, but by that same notion, we have no problem just staying at home.
With anyone else, they might be uncomfortable in this kind of environment. But it never gets that way with us.
So yeah, I'd say our friendship is different than most NT friendships. I think it's a very good one, though, and I'm really thankful that I've met someone like her.
As for other Aspies... I believe my cousin is probably on the spectrum. She had some difficulties when she was born and, for a long time, she would only be able to communicate with animal noises and body language (Non-verbal learning disorder, perhaps). She's now graduating with a physics degree. Needless to say, she and I get along very well and coexist well too.
While I can only speak for myself, I'm pretty sure I get along best with people who aren't what the world would call "normal". These are the folks I typically seek out (consciously or not) when I need some companionship. 