They are not different phrases. "Annoy to no end" is the American English version while "annoy no end" is the British. Both are acceptable and mean the same thing.

That's ironic, because the "no end" form was already a relatively new idiom. But "to no end" has already overtaken it in terms of popularity. Of course I meant the word "ironic" like I mean the word "literally"; which is to say, figuratively.

Being a prescriptivist creates no end of everyday pains. Language just won't conform.