Oh, dear, one of these questions that hopes for a logical answer. You've undoubtedly noticed that there's a lot of overlap.
Off the top of my head, I might say
(musical) genre = style of music, kind of music
What style of music do you like? Rock, rap, jazz, classical, world music?
Mozart composed in many musical genres, such as concertos, symphonies, and operas.
musical style = how a particular person or ensemble plays or sings, in a more descriptive sense
Frank Sinatra had a distinctive musical style because he shaped each phrase so carefully.
In order to perform baroque music in an authentic way, musicians need to be familiar with the musical style of the period.
I don't think I would say 'music style' or 'music genre' very often myself. In some cases it is more idiomatic in English to use a noun as an adjective than to use an adjective form, but I don't think this is one of them. Or at least, it sounds less common to me without -al. But it's not wrong, and you might see it in, for example, a list, where space is at a premium.