Civilization – When Capturing City-States do bonuses remain

Military will periodically gift you units. Cultural give you a boost to your per turn culture accumulation. Maritime gives you extra food at your capital, and small bonuses elsewhere.

Which military units you get depends on the age you're in, as does the amount of the bonuses for the other two, so there is a benefit to getting a tech from a new age to increase the bonuses. Being Allied instead of just friendly will also increase the size of the bonus or the frequency with which you are given units.

Its also worth mentioning that regardless of the type of city-state, if you are allied with them, they will give you any luxury or strategic resources that are within their borders. This can be useful to trigger "We love the king" days in your cities to increase the growth rate. When cities demand a resource you don't have, you can use the Diplomacy summary screen to see which city-states have that resource and then consider boosting your rep with them.

1. They can be significant threats, especially if there are enough of them against you. The game always ensures that their units are upgraded to the same level as the player with the highest tech advantage, and they are able to produce units pretty regularly.
2. They will leave their borders and are even capable of taking over enemy cities, although such an occurrence is infrequent.
3. Nation states can get to a point where they permanently declare war on you. This makes the diplomatic victory a lot more difficult since you can't rely on their votes. You also lose out on resource trading potential, which can be quite profitable.