I don't play Civ games anymore. Used to until TotalWar bursted on the scene with original Medieval (I skipped their first release based in feudal Japan). When Civ IV was released I tried to like it but could not really get into it. Something was always missing to me. And every time I launched either Rome or Medieval 2 I knew what I was missing. The ability to command your troops on the battlefield - decide your own fate even if the odds are against you, leave nothing to the turn of the dice as Civ does.
Civilization was a revolutionary game of the 90-ies but to me it never really improved from Civ 2 - just got overly complicated. TotalWar series showed improvement with each release, keeping me interested through the past 5 years. Both Rome and Medieval 2 have very active fan community with lots of mods, serving both to improve upon the official game and expand it into new areas. Also latest Sega release of Kingdoms Expansion Pack brings four new fully fleshed out and exciting expansions to the great franchise.
I am quite surprised you did not play either of the game. What planet are you from? Seriously!
But hey, you can buy both and enjoy them. They are under $25 on the used market, so the initial investment would not be that big anyway. After you find which game you prefer, you can try fan mods or go and buy official expansions. Don't buy expansions until you settle on which game you want to invest into, as they are real time whores.
For TotalWar there is also a great compilation available, called Total War Eras - including all three original games with their official expansions and lots of extras for cool $25. So check it out as well, as I can guarantee you that at least Rome is not dated at all and original Medieval is still a pleasure to play in custom battles.
Regarding your hardware - it will be a tough call for Medieval 2. It does require a quality graphics card or things can get choppy and outright unpleasant in battles. Even playing Rome on it would require you to make certain sacrifices in features on the battlefield. Best call would be to place pointed question on one of the Total War fan community boards (
http://www.twcenter.net or
http://totalwar.org/) and see what the responses will be.
Cheers,
netscorer.