A basement in the US can be for habitation or just storage of excess junk; in our house it contains the water heater, softener, heater, etc. etc. but will also be some type of big family room--haven't quite decided yet. We were just going to have a crawl space (about 4 ft. high, for purposes of accessing pipes, wires, if necessary) but then discovered the cost of making it 8 ft. high instead of 4 was minimal compared to doubling our living space.
In our area of the US, a "cellar" usually referrs to just a room under the house, often used just to store vegetables, or take shelter from storms.
Traditionally the cellar in the UK also had a coal hole where the coalman would drop the coal down either loose or in sacks. Lots of people therefore referred to is as the coal cellar.