- Impassible to land units and cannot be worked by citizens.
- Provides +1 Appeal to nearby tiles
- Provides Adjacency bonuses: +1 Faith to adjacent Holy Sites and +1 Science to adjacent Campuses.
- Serve as a source of Fresh Water via Aqueducts.
Note that the game has different types of mountains depending on the base terrain, but from a game mechanic standpoint they are functionally identical.
Mountains are valuable terrain features, as not only do they provide a defensive bulwark but also provide adjacency bonuses to the Campus and Holy Site districts. Building them in "valleys" surrounded by Mountains may boost their performance right from the start.
But even more importantly, Mountains may be used as a source of Fresh Water with the Aqueduct District! You just need to found your city close enough (within 2 tiles of a Mountain), and as soon as you manage to build the Aqueduct it will enjoy the full Housing bonus provided by bodies of water such as Rivers and Lakes.
As far as history is concerned, mountains are inhospitable obstacles to movement, and provide little to no benefit to a civilization ... other than as barriers against invasion. Thus, they are appealing only to photographers, hermits, and mountain-climber sorts.