Complement, supplement are comparable both as nouns meaning one thing that makes up for a want or deficiency in another thing and as verbs meaning to supply what is needed to make up for such a want or deficiency.
Complement implies a completing; it may suggest such a relation between two things or two groups of things that if they are put together they form a whole, or the full number, amount, or quantity necessary for a given purpose; thus, a grammatical complement is a word or phrase which must be added to a predicate if the latter is to make a definite assertion (as well in "he feels well," free in "to set him free," of no use in "it proved of no use").
However, the term even more often suggests such disparity in two things that what is supplied by either one is lacking in the other, with the result that their actual or theoretical combination gives a completeness that constitutes or approaches perfection.
Supplement implies an addition to something relatively complete but capable of improvement, enrichment, or enhancement by such an addition; thus, a supplement to a newspaper (often, a "book supplement" or "literary supplement") is an additional section which enriches the character of the issue.
Sometimes, however, the term carries the implication of needless addition. Sometimes, on the other hand, it comes close to complement in suggesting essential differences or a need of combination if perfection is to be attainedz.