They both mean the same thing, though you would probably use "any"
"I don't have any ink in my printer.", "I don't have any idea."
In the first case, "any" is used because ink, like water cannot be quantified. The second, with "idea" is just custom, more or less.
In normal spoken English, you would most often hear "I don't have any..." rather than "I have no..." It has nothing to do with correctness, just day-to-day usage.