"Self-confident" isn't good usage. The word to use would be simply "confident."
Comparing "confident" with "self-assured" (used as adjectives, not in a phrase like "confident that..."), the difference, if any, is very subtle. To me, "self-assured" has more of a connotation of confidence in a social context, whereas "confident" would be more general.
He was a confident public speaker.
He was a self-assured public speaker.
He was a confident skier.
Not: He was a self-assured skier. (You could say this, but it wouldn't sound natural.)