What's the difference between bessern and verbessern?
Is the earlier used mainly for health improvements (judging by the expression Gute Besserung!) and the latter for skill improvements (judging by Ich muss viel verbessern)?
bessern is chiefly abstract, about character of people and perhaps abstract Ideas.
Der Schüler muss sich bessern.
verbessern is about discrete qualities of things, generally so if the root is an adjective (cp "vergolden", "verletzen").
Die Situation, die Schulnote, die Gesundheit müssen sich bessern. Ich muss meine Noten verbessern.
The difference is not strict, because ver- is also used with verbs, and it might appear as if verbessern derives from bessern, so verbessern may be used in place of bessern. On the other hand, verbessern might appear like a reduction from verbessern, or a verbalization of Proto-Germanic *bataz "good", so they are used somewhat interchangeably. But if you act on something, it's strictly verbessern. That's the only difference I see.
You can see
bessern as the base form of
verbessern. The prefix
ver- adds a specification to the verb.
In this case it denotes that an action of a subject is taken to make something better, instead of it just happening.
In current language,
bessern is rarely used outside of idioms (such as
Gute Besserung) and
verbessern is used as a replacement in spoken language, although in my opinion the exact meaning of it got a bit lost.
The verb bessern is seldom used because it's about the lack of character of someone.
Er muss sich bessern.
That's something a person of authority would say about a delinquent.
Er hat sich gebessert.
That's what his defense attorney would say to the judge.
A century ago, people still said
Mein Zustand hat sich sehr gebessert.
to say their health condition got better. But nowadays that's
Mein Zustand hat sich sehr verbessert.
is a fixed expression as a vestige of that old-tyme language.
In almost all other cases, verbessern is used.