The differences between a mountain and a hill are difficult and nebulous. There isn’t a standard definition that sharply delineates one from the other. Even the United States Geological Survey (USGS) concludes that these terms don’t have technical definitions and no scientific consensus exists to determine if a person is looking at, standing on, or merely regarding a mountain or a hill.
In the UK, there used to be a standard method for defining a mountain and a hill. This was based on height, and for a while the US adopted this standard too. Both countries defined a mountain as being 1,000 feet (304.8 m) or more tall. Any similar landform lower than this height was considered a hill.
The US attempted to not just measure by height, but suggest that there had to be a drop of 1,000 feet or an 1,000 foot relief in order for a mountain to be considered such. These definitions were discarded in the 1920s, leaving us with no clear distinction between a mountain and a hill. It’s as vague as possible, though most people would agree that exceptionally tall mountains are indeed not hills. For instance, a person probably wouldn’t think of defining Mt. Everest as anything but a mountain, and certainly wouldn’t expect it to be considered a hill.
Most people would agree that significantly high areas with large drops are mountains, particularly when their names are preceded by the term Mount. Hills are more vague and are those softer, more easily climbed, areas of rising ground. Climbing a mountain to its peak or summit is called summiting, but people will rarely hear others discuss how they summited a local hill, unless they are in jest. Sometimes the difference is that mountains are named while hills may not be. Instead of relying on heights or drops, the definition in this way depends on local agreement that something is a mountain and deserves a name.
There is a film that references the old UK definition of the difference between a mountain and a hill. The film is The Englishman Who Went up a Hill But Came down a Mountain, made in 1995. The movie is actually based on real events in Wales, though the names have been changed. The basic events concern a certain town’s claim to the first mountain in Wales, named Ffynnon Garw. When a surveyor, played by Hugh Grant, comes to measure the “mountain,” he finds it short by about 16 feet (4.87 m) and declares to the town that they don’t have a mountain. The Welsh villagers respond by adding 16 feet of dirt, rocks, and various other things so they can keep their pride and the title of mountain.