Short answer: The West wins by a landslide. But the geological reasons tell a continent-scale story.
A near-continuous chain of ranges runs from Alaska to Mexico:
Tectonics drive it all: the Pacific Plate subducting under North America, plus Basin and Range extension.
Appalachian Mountains – Ancient (480+ million years), eroded, lower. Highest: Mt. Mitchell (6,684 ft). Once Himalayan-scale; now rounded, forested ridges.
Adirondacks (NY) – Not Appalachian; a dome of ancient rock uplifted recently (geologically).
Ozarks (MO/AR), Ouachitas (AR/OK) – Interior highlands, distinct from Appalachians.
Western North America sits on an active plate boundary (subduction, transform, extension). Eastern North America is on the stable interior (craton) of the plate—no mountain-building forces for 200+ million years. The Appalachians are fossils of past collisions (Taconic, Acadian, Alleghenian orogenies).
Result: The West has young, high, rugged, seismically active mountains. The East has old, low, rounded, stable highlands. Both beautiful—geologically, they're different planets.