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Cottages grow to provide more commerce

when worked over time, growing to town. You pretty much want as many science and commerce multipliers as you can get as the game progresses (almost regardless of the type of victory). Every town is +4 base per turn (or more if financial). Few other improvements beat that. Working lots of cottages in lots of cities means faster tech progression.
On the other hand, a different civic combo can give you another +3 science for each specialist you have and faster Great person (GP) generation, so the Specialist economy (lots of food to support lots of specialists in lots of cities) helps speed up your economy as well. GP are cheap early, so you can use them for big jumps early, but the GP benefit slows down as the game progresses and they get more expensive to generate. Specialists under the right civic, and a few special improvements can beat that kind of improvement to overall production (gold mine comes to mind) up to a point.
Back to cottages. Under the right civic combo that comes later in the game, you can get another +3

per town once you have matured your cottages to towns, so if you are still in a tech race in the second half of the tech tree, building tons of cottages early can keep you very competetive late game.
Once you learn the different ways to improve your economy in a general sense, you will learn that by specializing your economy (science city, production city, gold city) you can concentrate the science, commerce, and production multipliers in the dedicated cities for even more of a boost to your economy. For example, a science city will get an academy, a library, and as many science buildings and science specialists you can fit into it. A unit production city will get the forge, factory, and use engineer specialists, reducing your need to keep unit production going in science/gold cities (you moved population in the science city to work more valuable science specialists jobs so your production will be slower there).
Specialists and cottages are both useful, so most people seem to end up with what is called the Hybrid Economy. Cottage the capital, establish the optimal science city, cottage commerce cities, push units out of a production city.