It's exactly the same distinction as between "he" and "him", which for some reason are far less often misused.

At some point it became in vogue to use ``I'' everywhere. This may or may not have been due to the fact that people formerly ended up getting corrected from misusing ``me'', but not misusing ``I'', so it may have seemed safer to use the one that seemed more often correct.

Note that the formal ``do-er'' vs. ``done-to'' nomenclature is, respectively, ``nominative case'' (or, less commonly, ``subjective case'') and ``objective case''. This usually means that whatever comes before the verb is nominative and whatever comes after is objective, but there is one prime example where this is not true. The verb ``to be'' is always intransitive, which means that it does not take an object. That means that a noun or pronoun that is after the verb is not a direct object (done-to thing), but, rather, a predicate nominative (something that renames or recasts the subject). This means that the correct sentence isn't ``It's me'' but ``It's I''. I'm sure that there are some other examples that fit this (inclusing the trivial exercises of replacing ``I'' and ``me'' with ``he'' and ``him'', ``she'' and ``her'', or ``they'' and ``them''.), but this is, by far, the most common example.

Also, back to more normal rules, the object of any preposition is in objective case. That is things are done ``to me'', ``through me'', ``above me'', ``about me'', ``inbetween me'', etc., and never ``through I'', etc. But most people get that one right.