I ran across the manga first, then started watching the anime, and now follow both. They each have their own qualities that make them wonderful but I prefer the manga.
I've seen a few people in different places say they feel like the series is too repetitive but I honestly don't feel that way. I think there continues to be growth with each story-- most of the time it's a subtle character development but other times it's a changing relationship. And the times where there isn't a significant growth in a character or relationship, there's a meaning to the story.
I've also seen a few times people mention that they wish Natsume would learn other moves aside from punching and, yeah, there are a few times it looks like he's ready to just take whatever attack before someone saves him, but this doesn't bother me at all. He learned particular moves out of necessity as he grew up. Then when the series has started, he has a bodyguard and other people who will be there to support him. He does take care of a number of issues completely on his own, and the times he doesn't he probably either knows someone will help him, is not fast enough to be able to respond to sudden attacks (which is fair, since Ayakashi can be much faster than humans), or perhaps he even is willing to accept the attack because he's just that kind of person. (Although the slower attacks is one thing I think the anime could improve upon because sometimes the timing of it makes it seem like there's this long dramatic lag, which is normal for anime honestly since they like to up the drama, but doesn't necessarily seem supported by the manga.)
He looks to helping others and there are times he would let himself suffer if he thinks it will help others. I think he never really learned a sense of self-worth as he grew up, and that sort of thing takes a really long time to overcome. Every day is a struggle to keep holding onto the new life he's gained, and move further from the old life he left.
To be honest, I actually also don't agree that this series requires some sort of inter-human relationship as its endgame or for further development. One of the things I love about this series is it doesn't actually have romantic relationships. There are some relationships a person could read into if they wanted, but even then it's more like reading into it wanting to believe something could develop. This isn't a typical (I think annoying) series that relies so heavily on contrived sexual or romantic relationships. Everything about it is a step above/away from that. The point is much more subtle and because of that I think it's much more poignant.
As others have said, the series is heartwarming, heartwrenching, bittersweet, beautiful...
I really, really love this series, and I generally don't get into the slow-moving series because I oftentimes seem to have a short attention span for them. I didn't have any expectations the first time I picked up the manga, but I was pleasantly surprised. I've since started trying to get other people to read this series as well.
The anime strangely enough moves the manga around and randomly keeps a character around for no apparent reason, so for those reasons I prefer the manga, but the anime is also absolutely beautiful.
It's a quiet, mellow series that at times has brought tears to my eyes. I did find myself starting to fall asleep to a few episodes once, but that was when it was very late, I was very tired, and it wasn't at all because I was bored-- it's because the series is soothing. In fact, as cheesy as it sounds, I would go so far as to say the series soothes the soul.