Both Byzantium and Mordred are co-operative/competitive games. You are ostensibly playing knights of the round table and combatants in the 7th century fight between Byzantines and Arab Caliphs, but each game has only one winner.
In Mordred, you can be "greedy" and that tends to help Mordred out, but if Mordred wins then the person who helped Mordred least wins. Normally the greediest person wins. I haven't been able to get into the flow of Mordred, after a couple of plays. The game wants to be a quick 30 minute game, but every time the game seems like it will end people who feel they'd lose, work to prolong the game so they can win. I'm still learning though so it's not a knock on the game system.
I seem to be very good at setting my "good" knight moves for Mordred to attack (under the guidance of my fellow players), but somehow I won one game by building lots of cheap towns (instead of the more valuable and durable villages). Placing Mordred's forces seems to be the main source of player interaction in Mordred. The game is fun to play.
Byzantium, on the other hand, feels like it's rushing to an ending. Cities besieged until they are little more than piles of rubble fought over, and resources get more scarce. I still have no idea how to play a "good" game of Byzantium. There are a lot of subtle mechanisms in the game. After every play I realize how moves at the beginning of the game made the end harder for me because I wasn't strategically planning that far in advance.
I love how Byzantium makes you think about the resource management of a large scale military campaign. No simple tracing of supply routes with food automatically appearing for free if it could reach troops. Admittedly the system is highly abstracted, but poor resource management will hurt your score. Since troops cannot be disbanded the game rewards you for killing most of your troops in battle (so you don't need to support them from turn to turn). Although culling your troops means that you'll need to recruit more next turn (which might not really be cost effective, one of the subtle touches I still don't get yet).
Part of the subtle nature of Byzantium is the possibility of Constantinople's fall. I've pulled it off once, but that was probably caused by the other players inexperience instead of any skill on my part. Since you play both sides, you can leave Constantinople's defenses open (by taking control of Byzantium navy and the city's levies). However, players can still try to rush their armies to defend the city if they notice you lowering defenses.
Byzantium is the type of game (involves strategic thought and longer than an hour) game which I really enjoy playing. There is enough theme (desert transport, slight asymmetry between Arab and Byzantine forces) for me. Historical flavour, but not simulation (simulation really only works at the tactical level anyway, I want to change history not relive it).