what situation are they used in?

Something which is awkward is difficult to deal with, or ungainly, or clumsy. So saying that "he's an awkward customer" means that someone is difficult to deal with, or that the person himself is clumsy or graceless.

"He's a pain in the neck" means that he is annoying--he's difficult to deal with because he causes unnecessary problems.

Thank you, Setanta!

what situation are they used in?
I suspect that the "awkward customer" sentence is more British English; "pain in the neck" is American.