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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.

My opinion: People use 'don't like' more than 'dislike' because 'don't like' is a softer way to convey an aversion than dislike. Moreover, we should use the exact word that describes our feeling. If you use dislike in place of 'don't like', you will be left with nothing to express yourself when something that you 'dislike', happens.

Update: Happen to find another similar question.

Dislike is stronger than don't like. Don't like is passive; it's an absence of liking. You could for instance be neutral or disinterested. Dislike is active. It means you expressly do not like it and are therefore by definition not neutral about it.

If you imagine a scale going from dislike at -1 to like at 1, "don't like" is in the middle at zero. Or at the very least it refers to the entire part of the scale between -1 and 0, inclusive.

It depends on the context. "Don't like" can be interpreted in the same way as "dislike" e.g. "I don't like ice cream" is the same as "I dislike ice cream". They both mean I don't like any kind of ice cream.

In other cases, the two can have very different meanings e.g. "I don't like all flavours of ice cream" vs "I dislike all flavours of ice cream". Here, by using "don't like" I imply that I like most flavours of ice cream but there are a few that I don't. On the other hand using "Dislike", tells you, as with the first example that there are no flavours of ice cream I enjoy.

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.

The only non-obsolete meaning that the OED finds for "dislike" is "not to like," so I think it's safe to say there's no difference.

Imagine it like so:

Word                                | Emotion as number         | Example

------------------------------------|----------------------------------------------------------------------------------------

Love                                |  2  (max. positive value) | I love it!

Like                                |  1                        | I like it

(none)                              |  0                        | No comment / It's ok / I've no feeling towards it / 50-50 / etc

Dislike                             | -1                        | I dislike it

Hate                                | -2  (max. negative value) | I hate it!

Notice none of the 4 words above (far left column) are negated. Negation happens by adding "don't" before the word. Examples:

• I don't love it.


• I don't like it.


• I don't dislike it.


• I don't hate it.

When you use negation, it doesn't necessarily mean the opposite, it just mean it could be ANY other feeling.

• 
  I don't like it. actual emotion could be "I love it!".


• 
  I don't like it. actual emotion could be "It's ok".


• 
  I don't like it. actual emotion could be "I dislike it".


• 
  I don't like it. actual emotion could be "I hate it!".

Words in any language depends on context, so the meaning of a word may change, and also they way we say/express it.

Range:

I don't like it, but I don't hate it is a range expression which is between no-comment and dislike. So rating this feeling in mathematical equation could be like so:

-2 < Emotion < 1

So could be -1.9 ... 0.9

Answering the Question (Dislike vs. Don't Like):

Dislike = -1

Don't Like = ANY number that's not 1 (could be 2, 0, -1, -2), so it's ambiguous statement, which could be clearer by context + the way it's said/expressed/sounded.