I was thinking it might be altering an original versus modifying a method and creating something new, but it doesn't really work out: it seems both can be used for either. I don't see any difference. Are they exact/perfect synonyms?
Some of the comments notwithstanding, these are in fact synonyms, both with the meaning of "change." If you look up either word in a thesaurus, you will find the other listed as a synonym.
However, as with most synonyms, they do have a different shade of meaning, and there are some idiomatic phrases which exclusively use one or the other. As Lambie mentions, making alterations to clothes means sewing (or re-sewing) them to fit a particular person. For example, if you gain weight and have your clothes let out, that is an alteration. (If you lose weight, you would have your clothes taken in, which is also an alteration.)
Modification is generally a rather smaller change than alteration. For example, a modification to a will implies a rather smaller change than an alteration:
George altered his will to disinherit his oldest son.
George modified his will to leave his car to his niece.
As one of the dictionary definitions says:
to change somewhat the form or qualities of; alter partially; amend: to modify a contract.
It is for this reason that we would more often speak of incremental modifications to something rather than incremental alterations, and we would more often (although not as much more often) speak of wholesale alterations of something than wholesale modifications. But there are plenty of situations in which the degree of change isn't important where the words are interchangeable. For example, these sentences have pretty much the same meaning:
We changed our plans for the evening.
We modified our plans for the evening.
We altered our plans for the evening.