Don’t know either real well, just as a tourist doing recent stays for vacations and/or from passing through by motor vehicle to go skiing. In many ways they seem pretty similar, satellite communities near major ski resorts with year-round recreational/leisure activities, same general climate. Here are a few pros and cons:
Edwards: under Vail sphere of influence with more resorts/terrain on one season pass, busier ski slopes although Beaver Creek is pretty chill, more cruising terrain, closer to I70 and DIA (~130 mi), very close to Eagle airport, elev 7200’, pop 10,000, median house income ~$70k.
Basalt: under Aspen sphere of influence with less but not insignificant resorts/terrain on one pass, less busy ski slopes, more double black diamond terrain, further from DIA (~200 mi), very close to Aspen airport, elev 6600’, pop 4,000, median house income ~$70k.
I am more familiar with Edwards/Avon area than Basalt and dined and snooped around there. Edwards seems like a chunk of American suburbia plunked next to some fine ski areas. Since I am a suburbanite I am comfortable with that environment. That could be a con for some folks looking for a more mtn/ranch environment like Basalt. I didn’t get past the Whole Foods Market next to Route 82 to see the actual town of Basalt, so I don’t have a feel for it. I know nearby Carbondale a little bit and it seemed to have a more rural feel than Edwards. The proximity of Edwards to busy I70 could be a con for some folks, but a pro for others desiring quicker access to inexpensive flights out of DIA. If you lived at Basalt or Edwards you would likely be visiting nearby ski areas via your car and FYI after a lengthy stay five miles from Vail/Beaver Creek last winter I eventually found free parking with very convenient access to both Beaver Creek and Vail ski lifts. The year before I visited Aspen while staying in Carbondale and had to use free, but time consuming satellite lots and shuttle bus service to get to slopes except for one Sunday of free parking next to Aspen Mtn. But heard rumors of close free parking to Snowmass/Elk Camp. I think you could easily fall in love with either place. Does Edwards have a greater variety of real estate for sale including less expensive condos? If you planned to use your place primarily on holiday weeks then the lower crowds of Aspen might be a strong pro. If you planned to host many budget minded visitors then Edward's proximity to I70 and lower DIA airfares might be a pro. The variety of resorts in CO and elsewhere on the Epicpass is a pro in most folks minds.
PS: Forgot one pro for Beaver Creek with respect to older retirees, they have escalators;)