Seattle is more compact, smaller population, but a much more livable area with tons of young people living downtown. Berkeley is more edgy, more deliberately oddball, and the surrounding area to the south is run-down. You also have SF and Silicon Valley nearby with access to just about any cultural event you can imagine, plus lots of start-up people living in SF.
Pricewise, the bay area is a lot more expensive than Seattle. Both areas have excellent public transit and air connections. To me, Seattle feels a lot less crowded, cleaner and more relaxed. Traffic is horrible in both places. Both universities are close to the water, but Seattle’s transit system has ferries, the bay area uses bridges and tunnels.
If you are used to London, you will find food is both cities is excellent, though I’d give the edge to SF. Neither city has as many pubs, there are more coffee shop/cafes in Seattle especially.
You have lots of recreational activities in both places, In Seattle, a ferry connects you to Vancouver Island (Canada) and/or the Olympic Peninsula with some of the most spectacular scenery around. In Berkeley you can ski at Lake Tahoe a few hours (by car) away, go north to the Muir Woods and Pt. Reyes to hike and camp, or just hang out. It can be quite overcast in Seattle, but you also have lots of fog in the bay area.
For academics, no question that Berkeley (Cal) is superior to UW in just about every aspect except, perhaps , aerospace related fields. Cal is always boasting about their many Nobel prize winners ; UW not so much. Cal is pretty left wing, just right of Bolshevik I’d say, with lots of protests for no particular reason , it’s more of a tradition there.
To me, Seattle feels like a cohesive community, with traditions, distinct neighborhoods and history that its young inhabitants (from all over) have made their own, the bay area is more of a linked web. Seattle is a place to live; SF a place to work.
If you think of yourself looking back from 20 years in the future, would you rather remember the quintessential (for a European) California year abroad or something a little different, a little newer, a little more in tune with America today?