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."