Answer: In the case of, say, Bartok, none. Numerous composers have written music for teaching purposes in addition to writing "real" music. Kabalevsky is another. Even Shostakovich, he of the grand symphonic works, also wrote music for teaching. Bartok's Mikrokosmos collection for young pianists is as famous as his adult music.

There also have been and are composers who write only for students, or are known only for those works, but I don't think anyone will be able to find a psychological, morphological, physiological, or zoological difference between composers who write pedagogical works and those who do not.

Going back a bit, this tradition has a long and honorable history: Schumann's Album for the Young, Clementi's piano etudes... Chopin's piano etudes are a hybrid: they are teaching works, but also part of the standard performance repertoire. One of the first pedagogical composers is J.S. Bach, also known as possibly the greatest of serious composers. His Notebook for Anna Magdalena Bach was a collection of pedagogical pieces (not all by himself, to be sure) for his wife's use - these pieces have been played ever since by students who later perform his grander works.

But wait - this isn't really a theory question at all. What am I doing?