What is the difference between a House and a Cohort?

You join one or the other. A House is a player made guild, run by players. A Cohort is the game's own version of the same thing.

A Cohort is kind of made to be mercenary, but that is not what it ended up being used for. In practice it is where people go to chill, farm Territory War currency and never really do anything.

See also: House Finder if you are looking for a House, and Territory War and the seasonal system.