Chariots are complicated machines. They require technicians to build, animal husbandry to raise and handle the horses, a skilled driver to run the thing, and a warrior to fight from it. Also all the infrastructure to support all that stuff.

They were developed before we bred a horse large enough to carry a single warrior. Once we got horses that big, we can put one man on one horse instead of two horses carrying two men around. You can get rid of your technicians and don't have to worry about sourcing the materials for the chariot. You deal with half the horses, and they end up a lot more versitile.

That said, chariots hung on for quite a while. Caesar faced chariots in Britain in the 1st century BCE, and found them quite frustrating.

In addition to running around and throwing missiles, the Brits would use chariots to drop warriors off and pick them up when they got in trouble. We called them "battle taxis" in grad school, and that's how they're used in Homer. Caesar found that use both spectacular and incredibly frustrating, because he couldn't kill these guys on chariots. They would strike his line and run away. Nothing he could do about it.

BUT once you get horses large enough to that you get one man on one horse, everything about the logistics gets easier. The Parthians had that kind of horse at the same time Caesar was fighting chariots in Britain, and horse mounted archers absolutely massacred Crassus' army. You double your possible weapons pointed at the other guy just by getting rid of a 2 horse rig and the need for a driver. And then you can also ditch a lot of the infrastructure that you needed to build chariots.

Cavalry is just a better use of resources.