I should clarify by ancient warfare, I'm referring to Bronze Age warfare
First we should draw a distinction between light cavalry, a skirmishing force equipped with bows or javelins, and medium/heavy cavalry, which fights at close range with lances or swords. Horses were domesticated c. 3,000 BCE and were probably used in warfare soon after, but medium/heavy cavalry was a later innovation. It starts to appear sporadically in Late Iron Age and classical era warfare (Alexander's Companions are a notable example), but only became dominant in the Middle Ages. I haven't been able to find a convincing explanation of why. Some suggestions that I've seen, all disputed, are that effective medium/heavy cavalry relied on specially bred and trained warhorses; or stirrups and saddles; or advanced training and unit tactics. Whatever the reason, fighting hand-to-hand or charging on horseback doesn't appear to have been an option before at least the Iron Age.
So in the Bronze Age, the choice was chariots or light cavalry, and chariots were superior in a number of ways. They gave an archer or javelin thrower (usually two per chariot) a stable platform from which they could fire in any direction, whilst somebody else did the driving. Mounted archers, on the other hand, had to awkwardly maneuver a bow whilst trying to ride at the same time. The bows of the time were too long to swing over a rider's knees, so they could only fire on one side. Unlike cavalry, chariots could also be armoured and used for shock charges. Light cavalry was still used, especially by tribal raiders on the peripheries of the great empires, but proper armies used chariots.
Chariots also fitted into the social context of Bronze Age warfare better than cavalry. Horses were expensive, exotic animals. In an age when battle was idealised as the clash of noble warrior-heroes, it didn't make sense to waste them on a ragtag force of skirmishers that couldn't have a decisive impact in battle. Instead, they doubled down on the expense. Maintaining a fully equipped chariot team―the horses, the chariot itself, a trained crew―was ludicrously extravagant by ancient standards. It allowed for the creation of military elites in a similar way to knights in Medieval Europe. In the Near Eastern states, kings would either fund a standing force of professional charioteers, or else parcel out land or tax-receipts to vassals in return for them maintaining a chariot. In wartime this became the core of the army, with each chariot supported by infantry levies. Similarly, in tribal societies chariots were the preserve of a warrior-aristocracy, and chariot-riding chiefs would lead their retainers into battle.
The demise of the chariot came about from both technological advances and social change. The compact recurve bow and mass-produced bronze arrowheads both came into widespread use around 1000 BCE, making mounted archery much more practical. With the ability to shoot unhindered, a horse archer could now easily outmanoeuvre a chariot and the balance of power on the battlefield dramatically shifted. At the same time, military organisation was shifting away from the heroic clash of elites and towards well-trained soldiers acting as a unit. Generals in these newly professionalised armies preferred to have a drilled force of light cavalry that could conduct coordinated manoeuvres in support of the infantry core of the army, rather than a bunch of glory-seeking toffs careening around the battlefield in insanely expensive deathtraps.
Sources:
The response to this post by /u/FlyingChange may help answer the latter part of your question, but honestly it's not a very in-depth answer. I hope someone knowledgeable about this field of study makes a post, but from what cursory research I've done there seems to be ongoing debate about this subject.
Follow-on question, were there any cultures that "skipped" chariots and "only" used mounted cavalry?