Answer: Sometimes people use these terms interchangeably, but I think it's better to reserve the term "modulation" for a prepared key change, as opposed to an abrupt switch to a new key. Typically the modulation will involve a pivot chord that is found in both keys. For example, in C major one could write I, IV, V, I, ii, V, then alter the ii to a major triad or a major triad with a seventh. The new II7 chord now sounds like a V7 in the key of G, and the I that preceded it now seems like a IV in the new key - the I/IV is the pivot. Another method of modulation is to change the function of a diminished seventh chord: changing the names (but not the actual pitches) of some notes so that, for example, what was a vii dim 7 in the key of C becomes a vii dim 7 in a different key - the vii dim 7 is very flexible this way.

An abrupt key change is a device sometimes found in popular song, in which a verse might be repeated but with melody and harmony all moved up a step. The key changes, but it's not really a modulation. Think of this as being like rectangles and squares: not all rectangles are squares, but every square is a rectangle. Not all key changes involve modulation, but every modulation is a key change. Finally, not every change of key requires a change of key signature. A key signature is a convenience for reading since it cuts down on the number of accidentals, but many pieces change key without any change of signature.