Terms like "modernism", "post-modernism" and "futurism" are terms that are as much political as they are aesthetic. They are attempts to claim exclusive ownership of artistic validity. "We are modern, all you guys are just stuck in the past." Instead of making claim to a particular kind of approach, they simply condemn all other approaches. They are examples of emotive persuasion rather than arguments.
Now the situation is rather complex, because the ideology is usually separable from the art objects themselves, especially in music that tends to be ideologically ambiguous. The term "ideology" by the way, originated with the French Revolution as a coherent set of ideas and beliefs, not necessarily with a factual basis. The problem with the term "modernism" is that it is not only used as a weapon, but it is also rather non-specific in terms of what it refers to musically. Sometimes you get the feeling that it just means "what I like" or "what I dislike". It does clearly refer to innovation, but the ideological component is more than that: it not only praises a certain kind of innovation (serialism, for example) but it condemns other kinds of innovation and most particularly the desire not to follow the prescribed innovation. This pushes the term and practice from simply advocacy ("we all want to write serial music now because it is better") to political rigidity ("all composers must write serial music now because it is the only valid form of aesthetic expression"). This is to weaponize aesthetics, something that also originated with the French Revolution.
I said that the ideology was separable from the art objects. I am thinking of pieces like the Rite of Spring by Stravinsky for which he made no particular ideological claims. He simply locked himself away in a tiny room in Switzerland and came up with a new way of composing music--based, of course, on the colorful ballet music he had been composing, but developing and expanding the style into something truly innovative musically.
Much of the time the ideological component is stuck on afterwards by critics or others and bears little relation to the actual music. Take for example the term "minimalism" that has been applied to the music of Philip Glass and Steve Reich among others. It supposedly refers to music "that employs limited or minimal musical materials." But frankly, all pieces of music employ limited or minimal musical materials. By that I mean that it is a fundamental aesthetic principle in music that each composition have a specific focus on a very limited set of themes, rhythms, harmonies and so on. Compositions that do not follow this principle are diffuse, unfocussed and usually simply bad pieces of music. This is the first mistake that many student composers make.
What do I mean by "specific focus"? Whenever Bach wrote a prelude, he used an extremely limited set of materials:
So did Beethoven:
Both of these pieces use basically one rhythm and one arpeggio shape and simply vary it with harmony. This is quite minimal. This gives the music a focus that adds to its intensity.
On the other hand, the so-called "minimal" music of people like Steve Reich focuses in a somewhat different way, but is really no more "minimal" than a lot of other music:
The difference between this and, say, Bruckner, is what he chooses to focus on, what he chooses to repeat and what he chooses to vary. The first movement of Bruckner's Symphony No. 6 also focusses on a particular rhythmic pattern--a simpler one than Reich chooses--but develops it differently and instead of having a consistent pulse, has contrasting passages:
So what does the label "minimalism" really tell you? Not very much, it turns out. A better term, at least for Reich's music would be the one he prefers, "process music", or something more descriptive like "pulse music".
Just don't call it "modernism"!