MY 2 CENTS = composers and artists of the early 20th century that created the modern abstract styles, like starvinsky and picasso, were classicaly trained and their works reflect disciplined and well crafted creations, even through the abstractions - later composers and artists like schoenberg or warhol, usually never perfected their craft first and it shows in the haphazard and poor quality of craftsmanship - listening to a stravinsky work like the ebony concerto or petrushka is a joy but a schoenerg work, at least to me, is tedious and is hard on the nerves - guess its really a matter of taste like all music or art
Stravinsky's style was extremely different to Schoenberg's - Schoenberg experimented with various styles, including the abolishment of motivic relationships in Erwartung. Eventually Schoenberg's turn to dodecaphonic atonal music was imminent, a move that he saw nothing less than the 'elimination of the conscious will in art.' The controversy lies in the very way that Stravinsky and Schoenberg were being modern and especially in that Stravisnky did eventually adpot Schoenberg's twelve-tone technique, but ironically after Schoenberg's death!
I absolutely disagree with the previous answer, as it is solely based on opinion. Music is HIGHLY subjective, and especially when it deals with experimental techniques or entirely new systems, like the atonal system or serialism.
The comment about Schoenberg not perfecting his craft is completely WRONG. Anyone who has studied 12-tone music or atonal music knows that never in Schoenberg's music were a row of notes used that weren't carefully thought out and planned. Atonal music and especially serial music is written in a way that relies only on intervallic relationships between sets of notes. If you'd like more info, research set theory in music. A general overview: Every note of a C major scale is assigned a number (C=0, C#=1, D=2....). All enharmonics are the same number, so Db and C# would both be 1. If you analyze an atonal piece of music, you will find relationships among intervals and you will start seeing hidden patterns unlike any other style of music written.
The comment about Schoenberg's music being hard on the nerves is very opinion-based. Of course, it's not the type of music that you'd listen to at a party, etc, BUT it is appreciated for what it means to music and how music has changed since the days of Bach and Scarlatti. We have TONAL ears, nowadays, so the idea that atonal music should be enjoyable to listen to, is impossible for most folks. However, it may be difficult to listen to because it doesn't have any tendency notes or anything associated with tonal music.
There are many mistakes in the previous answer, but I can't go through them all. I doubt the poster knew even what they were typing.
ANYWAY, to answer the original question, Schoenberg and Stravinsky had heard of each others music, and even though Stravinsky began writing atonally, he thought Schoenberg's music was going in the opposite direction of what "music" should be going toward. Schoenberg thought the same of Stravinsky's style, he didn't think it was focused enough on motivic development of intervals and groups/sets of notes. (Stravinsky's style was more "primitive" (research the genre "primitivism") and fragmented as far as how different instruments worked together. Just listen to the Rite of Spring.
Schoenberg wrote a poem titled "Der neue Klassizismus", degrading Neoclassicism.
Anyway, that's just a dent in the corrections of the previous answer, and I hope I've offered some insight into the relationship between Stravinsky and Schoenberg.