I have played both Civilization 5 and Age of Empires extensively. Comparing the 2, these are some of the differences:
1. Tech tree
Civilization 5 and other 4X games have an extensive tech tree: you start with a single technology unlocked in the very first part of the game. You then have a choice of about half a dozen other techs, each of which can unlock other techs. Some techs require more than 1 base tech to unlock. Each tech has a specific theme that either gives you something new to use, or augments something you already can build.
Age of Empires has techs as well, but instead of a vast tree that spans the ages, it's more of a shrubbery: you build a new building and that building gives you the option to research a new tech. There are very few techs that require you to have 2 different buildings. Techs are also much more aimed at augmenting existing options rather than unlocking new options. They're mainly upgrades for existing units and buildings, rather than giving you an entirely new type of unit to play with or a new type of building.
4X games are on a far greater scale than traditional RTS games. Age of Empires is actually only a very small map when you look at it. if your map has a prebuilt city, this usually takes 1/3-1/4 of the map. Meanwhile, in Civ 5, you basically have an entire planet to explore, or a continent depending on the map type, and your city only takes a small fraction of that map, even if you include the culture region. Age of Empires only covers a fraction of human history compared to Civ 5: 1000 years in Age of Empires, at least 6000 years in Civ 5.
3. Advancement through the ages
4X games have a different take on entering a new age compared to AoE. In AoE, advancing through the ages is a choice you have to consciously make: you have only 1 town center for most of the early game, it can't build new workers while you're advancing, making it a meaningful choice that's not just "do it as fast as possible". However, it brings substantial long-term bonuses in terms of extra options. In Civ 5, advancing through the ages is more of a side effect of teching up: it's an indication of your progress and doesn't really bring anything new aside from some bonuses to certain currencies.
Diplomacy in 4X games, as other people have indicated, is far more involved than in Age of Empires. In AoE, you have 3 things you can do with other players on the map:
- You can change your disposition to someone between friendly, neutral and hostile;
- You can send them messages;
- You can send someone resources.
Anything beyond that isn't really supported and has to be negotiated via messages.
In Civ 5, you have an EXTENSIVE system of diplomacy, where there is a TON to do:
- Make several different military pacts, ranging from joint warfare against a 3rd player to open borders to a defensive pact.
- Send spies and/or diplomats to another player to steal their tech or improve relations.
- form the United Nations and vote on global proposals that affect all players and can have a major impact on progress.
- ask or even demand favors from the other player.
- when at war, broker a peace treaty with another player in return for some of their cities;
- trade strategic and luxury resources as well as gold.
- Establish trading routes over both land and sea.
In addition, there's a system of non-player city states that have their own disposition towards each player and can influence UN votes, give you extra resources and even gift you units.
5. City management
City management in Age of Empires is based around positioning your buildings in the right way. The focus is less on choosing what to build and more on where to build it and how much resources to dedicate to it.
Civ 5 allows you to micromanage each city and specialize cities for several purposes: generating gold/culture/science/faith, building units, getting a valuable resource before another player can,... The focus is on building an empire, not a city.
There are a ton of other differences, like how the win conditions are organized differently and how Civ 5 has a secondary tech tree based on culture. These are just some bigger differences that in general just indicate that Age of Empire is more a tactical game based on army composition, while Civ 5 is more of a management game where tactical choices take a backseat to empire management.