"Do you not" vs. "Don't you"

I live in the UK and I mostly hear people saying Don't you..., but some people say: Do you not...? What is the difference and which one is more correct?

You can put any example really. Something like:

Do you not like to come to school with me?
Don't you like to come to school with me?

Solution 1:

Both are correct. What was originally just a contraction of "do not" has become a word in itself, and can now be placed where the two separate words can't.

Both "Don't you..." and "Do you not..." are correct, but you can't re-expand "Don't you..." into "Do not you...".

The meaning of the two are the same, but "Do you not..." is considered more formal in some situations.