etwas wünschen is a polite way to say "would like" -- even more polite than möchten.
It's usually used with "you" as the subject, sometimes with "he, she, they", rarely with "I" since you usually use that kind of respectful language about others rather than about yourself.
sich etwas wünschen (with dative pronoun, e.g. ich wünsche mir) is "to wish for something" (e.g. as a birthday present, or to wish for peace on earth). That one can have any person as the subject.