Different strokes for different folks.
This isn't really a fair comparison. Empire Earth is a "real time strategy" (RTS), whereas Civ 3 is a turn based strategy (TBS). Empire Earth favors those who like more direct action and more direct control over wars, whereas Civ 3 caters to those who like diplomacy and dealing with things on a larger scale.
Its like comparing micro to macro. Empire Earth is micro, Civ 3 is macro. Civ 3 looks at the big picture. Sure they may cover the same time span (actually Empire Earth dwells into the future slightly), but the actions and wars in Civ 3 are more of a representation (each turn represents 2 to 10 years passing), whereas in Empire Earth, the action is real time, and there really is no measurement to how fast time passes (ages just seem to pass once you research the advance).
In Civ 3, one longbowmen unit is a representation of many longbowmen, whereas in Empire Earth each unit is exactly that, one unit.
Honestly I'm gonna get both games. They're the best of both worlds. If they could somehow merge the games, to include the deep diplomacy engine and macro scale strategy of Civ 3 and combine it with the direct control over warfare as Empire Earth, then you'd have one hell of a game. Unfortunately, this wouldn't work because the ways things occur in Civ 3, it just couldn't happen in real time, especially if you ever intend on finishing the game. For example, if you send in an airstrike to attack a civilization, it basically takes a year or two to perform in Civ 3. Each move represents a few years passing, how would you incorporate real time into that? If you had to deal with each war and action in real time, you'd never finish the game.
The truth is, I've always wanted to see some of the actions occuring in Civ 3. It just isn't as gratifying, winning a war without actually physically observing the attack. I'd like to see my bombers let it loose on the factories of my enemy, I'd like to see my longbowmen let loose a ton of arrows into my helpless enemies or 30 knights rush in and attack their archers. How do you combine large scale strategy with small scale strategy? Are you moving the armies from one continent to another? (Civ 3) Or are you asking them to travel a few metres around a few trees to flank the enemy? (Empire Earth) In Empire Earth you can pull some neat war strategies, like sending in your longbowmen, and upon seeing a rush of knights, you can pull them back, and lead them into a wall of pikemen. If you've ever seen movies like Braveheart or Gladiator, you can sense how exhilirating this kind of tactical warfare can really be, especially in real time.
With games like Age of Empires and Empire Earth, I've always wished for a little bit more diplomacy (like in Civ 3), unfortunately it wouldn't work because you couldn't very well stop in the middle of all the action in Empire Earth, and just start negotiating some kind of alliance, it just doesn't work. By the time you've discussed terms, you probably wouldn't have noticed that your economy is going to craps because you never controlled the villagers (you left them idle) and your enemy just ran over your city while you were negotiating. There just is no time in RTS, to stop and discuss diplomacy. It is strictly an "action game". That's the difference. It caters to those who like quicker action games. It depends more on reflexes & skill(how fast you can move and click the mouse properly) than actually strategy.
Just enjoy both games for what they have to offer.
(For the confused, contemporary American culture seems to regard them as such, even though WWI was about as as grey as it gets. And America did end up fighting against them, so that automatically makes them bad 
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