The only known way to prove that there's at least one prime in every arithmetic progression is by proving that there are infinitely many primes in every arithmetic progression. This is intuitively a fairly tremendous jump in difficulty to get the initial rather modest result out.
I imagine that most examples of this phenomenon take the form that the question as asked is "more difficult" only in the sense that it's been phrased in such a way as to mask what's "really going on." I think this is probably at the core of hundreds and thousands of problem-solving type puzzles -- the difficulty of the puzzle comes from masking the influence of the governing theorem, which is likely to be easier to see how to prove in its general form than it is to realize which parts of the puzzle are the important ones. In short, puzzles have red herrings, good theorems do not.
