They are really not comparable at all even though Age of Empires was heavily influenced by Civilization. Age of Empires is a classic RTS game with an "Age Up" mechanic, which is the main thing it took from Civ. Civ is THE 4X game: It's turn-based and it's more more on a macro/global scale.
So it kinda depends what you want. If you want to play an RTS, then yes, Age of Empires II is probably the best RTS ever made. Whereas if you want to play a good global-strategy game/4X game, then you can't really go wrong with Civ. I do prefer the older Civ games to the newer ones (Especially Civ II and Civ III).
Edit: An interesting evolutionary take on the Age of Empires formula can be seen in Rise of Nations, which leans a bit more into the 4X side of things. I think that they succeeded but the overall speed is lower and the controls and unit responsiveness could've been a bit less janky. Then you have Brian Reynolds' (who also worked on Civ) take on the Age of Empires formula with Empire Earth I & II, which is essentially: "Just add more stuff, man. Like, A LOT MORE STUFF, MAN."
On the Civ side, I think that particular formula was perfected with Alpha Centauri, which also allows you to do more with the environment itself. You can also take a look at the old SimTex games (Master of Orion/ Master of Magic) to get another spin on the classic 4X formula. There's also Colonization, which is a variation of the Civ formula but more down-scoped/focused on the conquest/colonization of the Americas. But I haven't played Colonization, since that whole setting doesn't interest me. Also: If you happen to like the Master of Orion games and you also like Dune and you'd want more complexity, try Emperor of the Fading Suns.