I'm a native english speaker. They both mean the same thing. "Do not" can be used more in a authoritative tone. You can use this on children or rules, so they can better understand the emphasis of a "no". "Don't" is just a shorter term and used more easily and loosely to shorten the sentence. 

The same goes for "You are" and You're" 

"We are" or "We're"

The first part is always more clearly stated. More so to get the point across.  I hoped I helped you. :)

'Don't' is NOT slang. It's standard neutral English - the contracted form of 'Do not'. It is the normal form to use when we are speaking. We use 'don't' in all situations when we are speaking - in both informal and formal contexts. The only time we say 'do not' is to add emphasis to what we are saying, for example to show that we are angry or that we want to contradict someone in a forceful way.

In written English:

Do not = formal

Don't = informal/neutral

In spoken English:

Don't = neutral

Do not = emphatic