Just as in logic, negation affects only the constituent it modifies, but in logic, negation can modify only sentences, whereas in English, it can modify other constituents as well. (A logician might prefer to call negation an "operator" rather than a "modifier".) The constituent modified by negation is called its "scope". Syntacticians test for where the scope of negation is by constructing examples with polarity items, positive and negative (see the Wikipedia article on polarity item).
As your intuitions probably suggest, in "dislike", negation modifies only the verb "like", but in "doesn't like ...", negation modifies the entire verb phrase. Accordingly, positive polarity "somewhat" is permissible after "dislike" because it is not within the scope of negation,
I dislike snails somewhat.
but negative polarity "at all" is not permissible,
*I dislike snails at all.
"Don't like" is opposite, since the entire verb phrase is the scope of the negation:
*I don't like snails somewhat.
I don't like snails at all.
The definitive syntacticians' reference to negation and scope is Larry Horn's A natural history of negation.