I've hiked in the Bernese Oberland many times over the years (first in 2005 and most recently in 2017) but never in the Rockies. Last month I remedied that by staying 8 nights in a cabin in Estes Park, CO with friends (all of us pushing 60). It might be interesting to discuss the merits and drawbacks of each from an American's perspective. We're all reasonably fit for our age, come from lowlands like coastal SC and Florida, and none of us are mountain goat types.
Natural Beauty
Purely subjective, but I give the edge to the Rockies. They are newer, more rugged, and about 30% taller than the Alps. With so much land set aside for RMNP, I saw much more wildlife on the trails than in the Alps. The drive from Estes Park to Grand Lake must rank among the top 10 in North America.
Hiking Trail Management
At RMNP we did 5 half-day hikes, one ranked as easy by AllTrails and 4 others considered to be moderate. Over the years in Alps, I have probably done 20 hikes based out of Murren, Wengen, or Grindlewald. All trails in both destinations were maintained well with good signage. I think I again give the edge to the Rockies because of the park rangers. They were Johnny-on-the-spot at trailheads with trail advice, geology knowledge, and, in one instance, elk antlers.
Transportation Infrastructure
The Alps win this category easily. To get to a trail in RMNP from the Denver airport, I had to take a shuttle from the airport to the Hertz rental car depot, and then drive 1.5 hours to Estes Park. From there, I had to drive into RMNP (and only during my strictly enforced timed entry window), then drive to the parking lot closest to the trail of interest. To get to a trailhead in the BO from the Zurich airport, I needed only take a train from within the airport itself, make one change with maybe a cable car ride thrown in, napping or admiring the views the whole way. Nobody does transportation better than the Swiss.
Culture
Another easy win for the Alps. The Alps are spiced up with cute towns, castles, munsters with Cezannes, Roman ruins, outdoor cafes, places where Hannibal crossed the mountains, museums dedicated to obscure patent clerks, and fields where the Swiss beat the Burgundians to secure their independence. Estes Park has a hotel that inspired Stephen King's The Shining.
Food & Coffee
I'm not a foodie so I won't hit this one hard. Restaurant food in Estes Park and the BO can be described the same way: decent but over-priced. Although Estes Park has an abundance of good coffee shops not named Starbucks. Depending on where you stay in the BO, you may have to go so some lengths to get to one.
Overall
All things considered I prefer visiting the Swiss Alps. One last intangible in its favor is that the Alps afford me the opportunity to get away from Americans. We're exhausting, especially during election season. Not sure I like other Americans very much any more.