Styles and schools might be the same concept; consider that in the 1900s there are no true schools, everyone has an autonomous and different creative path, but there are "affiliations," one might say.
The greatest school-non-school is Darmstadt. Many composers, especially in the '50s-'60s, qualify themselves as pro-Darmstadt or anti-Darmstadt, and Darmstadt means above all serialism.
The quintessential anti-Darmstadt is America, the United States, especially the so-called New York School, which is nothing more than the circle of friends of John Cage (Morton Feldman, Earle Brown, Christian Wolff...). And then minimalism, and Britten, and all the great ones still alive (Stravinsky dies in 1971, Shostakovich in 1975...), Messiaen (who is seen as one of the creators of serialism, but distances himself from it almost immediately).
If I had to analyze a composition from the second half of the 1900s and did not have specific indications, I would try to be very pragmatic. One must understand if it is tonal, non-tonal, modal, what the regulating principles are. It sounds like a triviality, but it is FUNDAMENTAL to understand and state it. Also because not all atonal-serial compositions are "truly" non-tonal... if in the mayhem you find a beautiful harmonic function, say so.
Then, macroform, microform, and instrumentation. Writing atonally is difficult; many composers have taken simple concepts and applied them to the entire composition. Most non-tonal compositions, especially those that are "generically atonal," are studies on an instrument, on a technical characteristic, or on a harmonic or melodic figuration. You won't find sonata forms, obviously; instead, you will find juxtaposed sound blocks, panel forms, very few repetitions. Try to understand the formal logic, even in a very simple way.