The dictionaries I have checked give translations for both plus and magis, and they seem to have a different tone. However, I have found no comparison between the two. They both mean "more" one way or another. I know plus is also a comparative adjective but I am interested in the adverbial use. What is the difference between these two adverbs? When is one correct and the other one wrong? Are there cases where they both work and mean the same thing? Or cases where both are grammatical but the meaning is different?
Magis is a comparative adverb (of magnopere?) meaning in higher degree, as seen in such expressions as magis magisque, more and more.
Plus is the comparative of the adjective multus, much, and usually refers to quantity, when it's used as if it were a neuter noun, as in plus plusque, more and more [of something]. It's also, of course, the comparative of multum, the adverb.
Each is used mainly to form comparatives for adjectives and adverbs that have none of their own, but not I think exclusively so. I was taught (a long time ago, since when fashions may well have changed) that they could be applied in the comparison of adjectives and adverbs instead of using a regular form : that is, that usage in prose composition can legitimately be subjective, style thus being allowed a part in the choice, especially where the writer is trying for an elegant effect.
I hope that this is sufficient to answer your concluding queries.