Welcome to the forums. ;)
Sadly, there's no way to increase the kill/death performance (through attack/defense values or otherwise) of units as they gain experience. While I love the game, this has been a sore point for me too as the dynamic, engaging feeling of watching my units gaining in strength through their victories can significantly boost my enjoyment of any strategy game.
The only somewhat field combat-relevant stat affected by experience is morale, which is handy, but it won't make your units fight better, just longer (as in, they'll need to suffer greater losses before they retreat - but they'll still die just as easily).
You can unlock stronger units however. There are two ways to do that:
- All factions in the game have a different selection of units they can recruit. Some of the factions have unique versions of certain units, like the Spartans and Thebans having Spartan and Theban hoplites respectively, which are larger (50-man) and have better stats than the average hoplite brigades.
Some factions also have unique units like the Persian cavalry (of the Persians, of course), or Tribal Archers (best ranged troops) from the northern tribal factions etc.
You can recruit these yourself from the cities of said factions as soon as you capture one.
- Some of the objectives in the campaign can also unlock special units which you can't recruit anywhere. There's only a few of these, but some of them are better than anything else in the game. Your Companion Cavalry will also increase in size through such objectives, though its combat stats will remain the same.
In the original Hegemony, the difficulty settings only affect AI aggressiveness as far as I know. They don't get any artificial bonuses (so the AI won't cheat), but (unlike in Gold) in the original game you can't run out of recruitable population, so the AI can pretty much keep you under constant attack if it wants to - and how much it wants to is what the difficulty settings control.