Unfortunately for you, this distinction is mostly “an ambiguity or a confusion of terms.”   In general (human-being ...) usage, these two words are pretty-much interchangeable.   In the Perl language, though, package is a keyword and module is not.

Therefore, just consider the idea:   Namely, that a great-big program can be subdivided into pieces, each one preceded by the package keyword, and that you can incorporate those pieces into other pieces by means of use.   That’s “the big picture.™”   Nevermind the terminology.   The language-designers, long ago, had to choose one word versus the other.