"Flat" is British English, and "apartment" is American English. That's the only difference.

'Flat' is British English, 'apartment' is American English. 'Apartment' is also understood in British English, and is sometimes used to describe a furnished flat which you might rent for a limited period, such as a holiday apartment, or a self-catering suite in a hotel. And while a 'flat' to an American is 'a shabby or low rent apartment' (see Dan Smith's answer), an 'apartment' to a GB.E speaker often suggests a spacious,stylish and luxurious dwelling. An interesting distinction!

None. "Flat" is common in the U.K. "Apartment" is the usual word in the United States. "Flat" is rare, but used and understood. Technically a "flat" means an apartment that is all on a single floor (as the vast majority of apartments are in the U.S.) and suggests a shabby or low-rent apartment. "Flat" in the U.S. might be a New York regionalism. A Damon Runyon story set in 1940s New York, written in colloquial English of the period, says"Dancing Dan first tries the knob, and right away the door opens, and we are in a little two- or three-room flat, with not much furniture in it, and what furniture there is is very poor."