It's the same thing really. It's just that Riesz first proved it for the special case and it was then generalized to Hilbert spaces. It's such a huge generalization that it causes a lot of confusion.
C([0,1]) is not a Hilbert space but a Banach space. Every Hilbert space is a Banach space, but not vice versa. The version of the theorem for Hilbert spaces is indeed a lot easier to prove than the one given in the article.