Why is there different climate types on either side of the Rockies, but not the Appalachians? Why does cities east of the Rockies have more precipitation than cities east of the Sierra Nevada?

Rain shadow zone.

Explanation:

Due to the circulation of the Bermuda high which creates a humid south wind that blows to the Mexican gulf along the terrain in east of the Rockies while the pacific high creates a dry northerly circulation over east of the Sierras. This results in the very dry region which is blocked by the Pacific moisture and blocked by the Gulf moisture forming a rain shadow over the Rockies.