Change, alter, vary, modify (and their corresponding nouns change, alteration, variation, modification) are comparable when denoting to make or become different (or when denoting a difference effected).
Change and alter are sometimes interchangeable; thus, conditions may change (or alter) for the better.
Change, however, usually implies either an essential difference, even a loss of identity, or the substitution of one thing for another while alter stresses difference in some particular respect (as in form or detail) without implying loss of identity.
Vary frequently implies a difference or a series of differences due to change (as by shifting, diversification, or growth). Sometimes it implies a deviation from the normal, the conventional, or the usual.
Modify suggests a difference that limits or restricts; thus, an adjective is said to modify a noun because it definitely reduces the range of application of that noun (as old in "old men" and red in "a red rose"). Often the word implies moderation (as of severity) or toning down (as of excess).
Modify may sometimes suggest minor changes or absence of radical changes.