Good question. I often see people saying "but that's not ancient, ancient refers to (usually the fall of Western Rome, or something like it)" which is partially true within a certain context. But the word can be used in other areas -- including histories -- in different contexts. For example, you might describe an Atari as an ancient video game system (to use a very extreme example) because in comparison to the PS5 it is. But you could not refer to an Atari videogame as an ancient game, because games themselves do go back into antiquity (note the usage there).

And while it's less common in modern scholarship, you can find works describing the ancient history of places like New Zealand, covering periods around 1300 CE because that's when we're getting the earliest human activity there. the Ancient Art sub had to try and figure out cut-off dates, and while I'm sure there's some work that could be done on it, they decided that this would depend on geographical location. https://www.reddit.com/r/ancient_art/comments/k62ml1/ancient_art_timelines_and_rules/

But with antiquity there's less wiggle room. When used on its own it usually refers to the regions around the Mediterranean up to the fall of the Western Roman Empire. You can use it in other specific examples, but you would need to be specific on where if you wanted to be understood. But it would stick out like a sore thumb if you used it to describe video games from antiquity, or claim that the Maori tribe had existed since antiquity. This even gets carried over in terms like antiquities, which have a fairly time sensitive meaning, or at least a stronger cut-off.

So while neither ancient nor antiquity are older, the term ancient has a much longer span and can be used in more contexts than antiquity. Hopefully I've made sense.

By definition, "ancient" means everything before the Middle Ages, commonly understood as everything before 500 CE. So that includes ancient Egypt, Sumer, the Hittites, China, Greece and Rome, etc. On the other hand, when people mention "antiquity," they're generally talking about the period of ancient history circa 800 BCE to 500 CE. This is defined as the period coinciding with the Greco-Roman period, but it's important to note that there were other important developments going on in other parts of the world during this time as well.