A basement is generally a full story below (or partly below) ground level that's°hard word designed as°hard word usable°hard word living or utility°hard word space — it typically°hard word has higher ceilings, may have windows, and is often finished or finishable°hard word. In modern°hard word North American construction°hard word, "basement" is the default°hard word term for any below-grade°hard word level of a house.
A cellar°hard word, on the other hand, traditionally°hard word refers°hard word to a below-ground°hard word space that's°hard word more utilitarian°hard word — lower ceilings, dirt°hard word or rough concrete floors, minimal°hard word lighting, and primarily°hard word used°hard word for storage°hard word (root cellars°hard word, wine cellars°hard word, coal cellars°hard word). It carries a connotation°hard word of being unfinished°hard word and not°hard word meant°hard word for habitation°hard word.
There is a regional°hard word/dialectal°hard word layer°hard word to it, though. In parts of the northeastern°hard word US and in British°hard word English°hard word, "cellar°hard word" is sometimes used°hard word interchangeably°hard word with "basement" regardless°hard word of the space's°hard word condition. Older homes in New England might have what anyone else would call a basement referred°hard word to as°hard word "the cellar°hard word" out of habit°hard word. In some building codes°hard word (notably°hard word New York°hard word City's°hard word), the distinction°hard word is actually formalized°hard word: a basement has at least half its height°hard word above curb°hard word level, while a cellar°hard word has more than half its height°hard word below — which matters for zoning°hard word and legal°hard word occupancy°hard word.
So it's°hard word partly regional°hard word habit°hard word, partly a real difference°hard word in the type of space being described°hard word, and occasionally°hard word a legal°hard word/technical°hard word distinction°hard word depending°hard word on where you are.