Well, you summed it up here:
I suppose I'm just very different and enjoy having a good level of intimacy with a few close friends.
Yes, you are different to her. Perhaps she can't be the friend you want, or doesn't want to be. Don't take it personally. Some people are perfectly happy being different to you, and it isn't a reflection.
Tell her how you feel but don't expect her to change. If you can't be friends with her how she is, and that's fine by the way, perhaps think about putting some distance between you two, because it sounds like it is more upsetting for you than it is for her.
On a personal level, I am that friend. Your opening post pretty much sums me up, actually. I have a few amazing friends who understand me and accept that that is how I am and we have a lovely time together and I love them lots. I have a couple of friends who I am intimate with, but to be honest, past one or two people, that stuff is SO draining. I have one friend who is pushing me for more and it feels like she is literally digging under my skin, like it's actually painful and exhausting and I can't cope
... I don't know how to tell her, so I keep her at an arms length. 
I accept that the way I am drives some people away, and I have lost friends because of it (mostly work friends wanting more than just work interactions) I can live with that, though, because I know that being someone else to keep them around makes me so utterly miserable