It's generally understood that the coordination conjunction 'but' is used to create contrast between different constituents (e.g. positive against negative, good against better, bad against worse). However, there are many examples where the contrast is less overt, particularly when 'but' is used to connect independent clauses.
I believe, though correct me if I'm wrong, that the contrast is created by subtext, an understanding held by the writer/speaker and thought to be understood by the reader/listener.
In the following clauses, what is the understood contrast between the coordinating elements? I've added my thoughts alongside each example:
[1] She lost a great deal of respect for him in that moment, but she would never admit it.
In 1, I can't see any obvious contrast between her losing respect and never admitting it. Are we to take it that the contrast stems from an inferred belief that she would admit it if she felt it?
[2] He stared at the mess, but he tried to hide his unease.
In 2, is it to be understood that him staring at the mess implies unease and that this is contrasted with him trying to appear as though he's not uneasy?
[3] He went the pub, but she wasn't there.
In 3, I believe there is the understood idea that he went to the pub to find/see her, which implies that he believes she is there, and this contrasts with her not being there.
[4] He started crying, but he pretended that he wasn't.
Example 4 is perhaps the most obvious contrast, but I've included it because it isn't as simple as 'He started crying, but he stopped.' He is still crying, so the only contrast can come from the fact that he appears not to be.
[5] Maybe I'm wrong, but correct me if I am.
This last example, a direct quote from Oprah, is interesting because the two clauses seem to agree with each other (they're both negative, that is, at the surface level). However, the use of 'but' almost implies that she thinks she isn't wrong and that the 'maybe' is her saying, 'I don't think I'm wrong.'