"I have no ..." vs. "I don't have ..."

The two formations are essentially identical in literal meaning, but the phrasing may be chosen for dramatic emphasis. Probably the most famous example of this construction is the cliched father who's disowned his child: I have no son! It literally means the same thing as I don't have a son, but it's much more emphatic.

Edit: The have no form, since it's technically correct but not used as often, is sometimes used for humorous effect to imply that English is not the speaker's first language. The most famous example is probably the 1922 novelty song Yes! We have no bananas.

Re-edit - updated YouTube link, as previous video had been taken down.

"No" in this context means zero.

"I have no house," means, "I have zero houses." "No," is therefore a measure of "quantity."

"Not," is just the usual negation. This is a "qualitative" statement.