I had an intuitive answer for this, and have done a bit of searching. There are various results, some of which agree with me, some don't.
My intuitive impression is that a basement forms part of a buildings foundations, and a cellar is merely a room. Also, we tend to think of cellar as a place for storage (a wine cellar, for example); and a basement as something that can be used as you wish: "I am going to turn the basement into a gym" for example.
However, some research suggests that you can use them interchangeably.
Possibly it is the case that there once was a distinct difference in meaning that has been eroded over time and now people use basement/cellar as if they were the same.
Sadly I have no definitive answer for me, but the most satisfactory answer I found from research was on quora:
"A basement is the lowest floor a building that is mostly or entirely below ground; it is typically part of (or at least occupies the space between and around) the structural foundation of the building.
A cellar is a room below ground level that is typically, but not always, either attached to or beneath the house it serves. This room is built for a specific purpose, and the name is often times a good indicator, such as with wine cellars, root cellars, or storm cellars.
The terms are often used interchangeably, so it's understandable that there could be some confusion. Often times a cellar can be part of a basement, so you could be considered as entering both of them at once by descending a flight of stairs or equivalent. The distinction is that a basement is an area below ground that is part of the building's foundation but doesn't have a specific purpose, and a cellar is a below ground level room that you are using for its cellar-like traits (temperature and shelter) but doesn't have to be part of the home's foundation."
You can see this matches, more or less, my intuitive answer.