I've seen both "супермаркет" and "гипермаркет" being used to describe what I would call a "supermarket", but what is the difference between them?
There is absolutely no difference, just pure marketing.
First, a historical context of how a word “супермаркет” appeared in the first place:
See, here is already such a word as “универсам” (УНИВЕРсальный магазин САМообслуживания — “a universal shop of self-service”). However, in 90's, when supermarkets really started to boom, this word was thrown away an a one of soviet origin and replaced with an English borrowing. The reasoning is simple: English is a language of the USA, and in 90's everything American was believed to be modern, forward and advanced. Also, you could have absolutely no knowledge of English, but if you used individual English nouns, you showed that you are cool and modern.
So after this, a word supermarket became a very effective advertising hook with a tint of a foreign origin, even if these supermarkets had absolutely nothing with foreign brands or even consumer standards.
However, time passes, every big shop becomes a supermarket and a need to stand out from competitors raises again. Since a layperson understands only a “super” part and has absolutely no clue about the second one, some marketing managers decided to replace “super-” with “hyper-” to sound cooler without getting any obligations.