In a nutshell I have a 'friend' who is in my life due to circumstance. We are very very different people. Some times when we catch up we will have a lovely time and I think that a real friendship is possibly developing, but then I will hear something or see something that reminds me quite clearly that we are not (I am embarrassed to even say these things but stuff like not adding me as a friend on FB, and not inviting me to her kitchen tea) These are things I would normally not care a jot about but when we see each other in the flesh she is very friendly and I think we are finally making headway, but obviously not. If it was just a question of her being from my friendship group, I would just walk away as it would be easier to not have someone in your life who makes you feel crappy. But this isn't the case and we will be seeing each other frequently over the years.
When we catch up I find myself babbling about myself and my life and giving away far too much info while she gives nothing - I am just desperate to fill the (uncomfortable) silences. This is soooo not like me!!
So my latest thinking is that I need to be the one to deal with this issue. It isn't the kind of situation where I could discuss the issue with her face to face (even though that is the type of person I am and the approach I like - get it all out on the table and be honest).
So:
1) am I a nutcase for even spending this much time thinking about it?
2) anyone have any experience managing a situation like this?
3) I feel like I need to become more mature and secure within myself, but how to do so? I am in my 30s now, so like Bridget Jones need to become sensible and mature

This driving me nuts so any advice would be appreciated!!