Is there a difference between "dislike" and "don't like"?

A friend of mine for whom English is a second language told me that I am the only person he knows who uses the word "dislike", and asked me what the difference was from saying that I "don't like" something. I answered that they were the same, but that "dislike" might be more formal... but it occurred to me that I was not totally confident with this answer.

Is there a difference between "I dislike that" and "I don't like that" besides the former being more formal?

Saying that you dislike something means you have a distaste for or hostility towards it.

When you don't like something, it means that you would prefer something else over it. You would want this to not happen.

Source: Apple's dictionary.

'Don't like' means that even though you would prefer the other thing you can still adjust to this happening. But saying that you 'dislike' means that you will be really unhappy if the thing that you disliked, happens.

Update: Happen to find another similar question.

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.