I largely agree with Umbran in that setting tends to be a strong factor, though magic is also a prominent one, in determining how high on the "scale" of fantasy something is.
I personally look at it as being measured on three axes:
The X-axis is the prevalence of magic. This covers how often you run into magic/fantastic elements, as well as how much people believe in them/are knowledgeable about them. If magic is rare, hidden, and not widely believed in, then there's a low mark here. If it's fairly well-known, and you can find a spellcaster as easily as you can look someone up in the phone book, then there's a high mark here.
The Y-axis is the power of magic. This is how strong the fantastic elements of the setting (which is really what "magic" is shorthand for) are in terms of their ability to accomplish things. Lord of the Rings gets a low mark here, because nobody is tossing out cure spells willy-nilly, dragons are exceptionally rare, and magic items are legendary lost treasures with names and histories. By contrast, the Forgotten Realms has a high mark, since magic can quite literally move mountains, dragons and demons and beholders are everywhere, and there are artifacts that can banish gods and freeze the entire planet.
The Z-axis measures the setting, in terms of its connection to Earth. The closer the setting gets to Earth (both in actuality and resemblance), the lower the score here, and vice versa. That sounds odd, since this tends to be thought of as binary - either something is set on Earth of it isn't - but fantasy is broad enough that this is more of a continuum. As the past is (sometimes quite literally, in fantasy) a different country, the scale here measures not just the connection to Earth, but also to contemporary Earth (though medieval Europe is such a standard now that it barely moves the proverbial needle if it's set there).
For example, as Umbran mentioned, the Conan stories get a low mark here, since they're set on what's explicitly Earth in the past, with the place names being different but not unrecognizeable. Harry Potter, by contrast, gets an even lower score, since it's set on modern-day Earth - common place-names abound - and has its goings-on simply hidden out of sight in small towns and castles. By contrast, Star Wars has a very high mark here, since it's not even set within the same galaxy.
The higher the marks, the closer something is to "high" fantasy, and vice versa. Of course, this leads to odd places when some axes have a high mark and others a low one, but that's more of a problem with the terminology than with the scales.