We use the two expressions slightly differently. Your car keeps breaking down and your friend says "why don't you buy a new one"? "Because I haven't got any money" - of course you probably do have some, but not enough to buy a car.
If you say "I don't have any money " it is more likely to mean I don't have any money with me right now (but you do have some money available - perhaps at home, or if you went to the bank).
Whichever version you use, you need the 'any' in there between 'have' and 'money' for it to sound natural.