Its hard to say what is best for the game, but I do know something about history. Of course, FfH is a fantasy game and we can change history.
Early Chariots
Chariot technology was developed in many places around the world in stone and bronze age civilizations. In the Iliad, we see chariots mostly used for transportation and somewhat for mobile attacks. The chariots would have a driver, and the non-driver then would throw a spear. He would then either dismount and fight on foot, or repeat the tactic. Note it then takes multiple people to launch an attack. Also, chariots were used to recover wounded on the field.
The Assyrians made major use of chariots in their conquests. Most of the early iron age civilizations could not control large armies of foot soldiers, and they were dominated by an elite class that would use chariots.
As many people have said, they were pretty much useless unless on flat ground and badly unmaneuverable. Most people like the Hittites and the Assyrians had large three man chariots. The Egyptians, however, had a combined arms force and used lighter two man chariots.
In our tiny knowledge of the battle of Kadesh, it seems that the Egyptian two man chariots proved to be superior as being more maneuverable. It is also interesting to note that the battle was mostly an arrow and chariot battle, there was not a lot of fighting by the foot soldiers. So, we already see, even as unwieldy as chariots are, that mobility in open spaces was critical.
At this point, there were few 'horsemen'. The Romans had a class of 'equities' who fought on horse, and even the Greeks had a few horsemen, but the only group using horsemen as their major force at that point were the steppe nomads, originally the Scythians (and then the Sarmatians). In the far east, there were other nomad horse riders. They were considered the best horsemen, fighting directly from horseback using bows.
However, at this point, chariots could be used as shock troops, while cavalry could only be used as a light force, with bows or as classic light cavalry.
One ominous note was the ancient dark ages. The Hittites and the Middle East were overrun by a group called the 'Sea Peoples'. Militarily, how did this group defeat the Hittite Chariots? Many historians feel that they were something rare in the ancient world, a real infantry army that was large and they potentially could defeat an elite army of chariots. The Egyptians defeated the Sea People, ironically, at sea.
Later Chariots
The Persians had a force of scythed chariots, that were their main shock force. Indeed, it appears that Darius III's biggest error was relying on these potentially devastating weapons against Alexander. We can imagine their power against the various groups bordering the huge Persian empire; but they were a total disaster against Alexander. Alexander had what we would view as professional troops. They simply opened up their lines as the scythed chariots just rolled through; once the chariots lost momentum, they were slaughtered by the Greeks.
The Celts in Britain also used chariots mostly for shock. Celtic warfare was often badly coordinated, and the chariots were useful since their inability to coordinate was not so much of a burden. However, against the disciplined Romans, there performed poorly.
The Rise of Cavalry
During Roman times, there were always powerful armies to the East. The Parthians occupied what we now think of as Persia; they were a Sarmatian tribe that had their roots in the steppes. The Romans always had the upper hand, slightly, but they had a tough time. In particular, the Battle of Carrhae was a disaster. It is interesting to see the usual Roman tactic of handling horse cavalry -- it was to go into Testudo (turtle) formation and wait for the enemy to run out of arrows!
In the early third century, the Romans had a new, greater problem -- the Parthians were overthrown by the Persians. Also skilled horsemen, they had a central taxation system, advanced culture, etc. Rome and Persia fought incessantly for the next few centuries even though it was clear neither could defeat the other.
The stirrup is usually given the credit for changing cavalry to the dominant military arm. Clearly, the stirrup was a critical invention. However, we must also recognize that between the stone age to the late iron age we had also breakthroughs in animal husbandry, care of horses, armor and shoes for horses, fletching and the testing of the best feathers, arrows, bows for shooting from horseback, cavalry tactics, and perhaps other improvements.
In the late fourth century, the 'key' battle of Hadrianople is often cited as the period that cavalry surpassed infantry, as the Gothic cavalry simply rode down the Roman infantry. Of course, these weren't the Romans of Caesar or Trajan, but cavalry was here to stay.
At this point, we do often see heavy differentiation of cavalry into archery cavalry, classic light cavalry, heavy cavalry, and even medium cavalry. The Byzantine army had cataphracts, who trained to fight on horseback but also dismount.
The absolute masters of cavalry warfare were the Mongols. Their coordination, combined arms, and tactics (including a series of dirty tricks) are almost unbelievable. By this time (12th century), we see all kinds of fancy innovations: silk shirts to reduce arrow damage, groups for care and feeding of extra horses and rotation, special arrows (they had armor piercing arrows, arrows for high shots, etc.), specialized fletcher's and arrow makers, etc.
Conclusions
Chariots were useful early for mobility, especially in periods of non-professional soldiers.
Early chariots were used mostly for light use, but more organized armies like the Assyrians used them for shock. The Persians used them this way.
Early cavalry was more maneuverable and therefore superior, although it was hard to fire bows from horseback.
Substantial technical innovations for cavalry had them advance as a force of warfare. Chariots are obsolete. The stirrup were a key development.
By the late 4th century, cavalry had risen to the dominant arm and a great variety of cavalry was available.
Cavalry flavors advance to specialization, and can be used for shock, archery, and light purposes and other specialization.
In the Game
I think regular BtS does a pretty good job for a simplistic game. Chariots are introduced early where they can be useful. They are supplanted by light cavalry and then knights.
The biggest problem I have with the use of cavalry in BtS is far more complex, namely that city combat is far more important and therefore cavalry appears less important perhaps than it should be. Also, with horseback riding as a dead end tech, it loses its appeal.
As far as FfH goes, I do like the idea of having a heavy and light track, or perhaps an archery vs. non-archery track and have the differentiation coming from upgrade choices. I also think that chariots should be an early game choice and true cavalry later.
Personally, I think that cavalry should be made stronger in the open; they should be tougher than the the scout-hunter line. Perhaps they should get a big bonus for fighting on grasslands or plains to offset their disadvantage in cities.
Best wishes,
Breunor