There is a difference.
'Don't make noise,' means exactly that: Do not do anything to create noise. Other, perhaps more common, phrases are, 'Don't be noisy,' or 'Please be quiet.' You might make this request to a group of rowdy children, or to someone having a too-loud conversation in a library or other 'quiet' place.
'Don't make a noise,' (or, more commonly, 'Don't make a sound') is something you would say if you were in a situation where it was important to be absolutely silent. Examples are: hiding, sneaking up on someone, trying not to wake someone, etc.
I hope this is helpful,