For a family vacation in June where you want to take it easy in a mountain ski town, would you go to Vail or Breckenridge?

NOt Vail. Breck would be fine. Aspen could be. Vail is a "man-made" destinaation, great for skiing but I'd say a bit soulless for family in the summer.
You can do a number of placess easily once you are in the mountains. Young folks might enjoy the Woodward facility at Copper Mountain where trick skiiers and snowboarders practice their tricks--lots of jumping into foam rubber pits. Or ther is zip lining or fly fishing.

We love Aspen, especially in the summertime. Lots of activities like fishing, hiking, river rafting, music events, etc.
Agree with Vail-- not our cup of tea anytime, really.

Breck.

Actually I think Vail kind of fun in the summer time. Great 4th of July parade. Large numbers of sports camps and other activities going on. Some great restaurants at discount prices, great sales on winter gear. single track biking, etc. Personally I am very tired of the phrase, " "man-made" destination, as if the other areas are not man made. Granted Breckenridge and Aspen are old historical mining towns but damn little of those towns still exist under the glitzification of downtown areas with high rise condos everywhere.

What is the age of the children? That might make some difference. Breakenridge has a lot of similar activities and would be my preference in general. But I would not automatically exclude Vail. It has it's features.

Everyone for your thoughts. The kids are 11 and 13

With kids that age, Breck is a better option. The would like the Alpine slide at Breck, there is a gold mine tour and a bit closer to the Georgetown Loop Railroad. Take one of the mine tour options. There is fake gold panning with one of the tours but it is fun and you will have a hundredth of a gram of gold out of the deal. If you are lucky, maybe a little more. There is Mineral Belt trail in Leadville. Twelve mile loop, hard surface - mostly flat - that goes deep into the old mining areas around Leadville.

On the way from Denver, there is great railroad museum in Golden.

Vail is 8,000 ft . Breckenridge 9,600 ft. Coming from sea level I have a hard time at Breckenridge.

Breckinridge. As to the altitude, I've had problems there w headaches. For first couple days, lay off the alcohol, keep yourself well hydrated, if near the summit, take it slow. Love that town. Vail was like a fake Zermatt IMHO.

all good considerations. Thanks.

Wrench in the works time: Winter Park has the mountain recreation and lots of lodging, and Lake Granby, Shadow Mountain Lake, Sulphur Hot Springs, and Rocky Mountain National Park are much closer as side trips.

That’s how we feel about Vail, too, aliced. It’s fake Swiss, stuck right next to the highway. Yech.

Lots of things to do in Grand County in the summer. We have gone there primarily since the 70s. I’m happy to answer questions about it, if the OP is interested.
All-time favorite: Devil’s Thumb Ranch, Tabernash, between Winter Park and Granby. Awesome family venue, gorgeous mountain setting, great food, stellar customer service, lots of special activities.
Close enough to day-trip to RMNP via Grand Lake over Trail Ridge road, or to Steamboat Springs.

Mind you, I am not a Vail hater. We spent many spring breaks there for skiing in Colorado and it made our kids love skiing AND Colorado. And summer in Colorado is as much a tourist draw as skiing in winter so there is a LOT to do. But lots of great places too.