Arguably, a few changes helped the cavalry replace the chariot.

The first is that horses were starting to be bred bigger and with combat in mind- Xenophon talks extensively about finding a good warhorse for riding in his text, "The Art of Horsemanship." He references a now lost text by Simon on charioteering, but he made it sound as if the chariot was already becoming obsolete.

Next, a chariot has some pretty major flaws. First of all, an old style chariot with wooden wheels and no suspension system would be bouncy as fuck. Get in a red flyer wagon and role down a hill, Calvin and Hobbs style. That's what the experience would be like.

When you are on the back of a horse, it's much easier to deal with rough terrain. Additionally, it allows you to traverse more types of terrain. You can ford rivers, go up and down hills, make tighter turns, and go through brush with much greater ease.

A chariot also requires more horses, so you have a greater risk of loss. It also increases the necessary logistics train to supply them.

Another key development is the saddle. While we don't know exactly when the saddle came into existence, there were proto-saddles in Central Asia and the Middle East circa 500BCE. The saddle allowed soldiers to carry equipment and sit on the horse with more comfort.

The Battle of Mons Graupius (~84CE) is the latest account of chariot use in warfare that I can find... and apparently they didn't do so well.