We prefer Alta over Snowbird for our now 12 and 9 year olds - east coasters, southern at that mind you, so minimal ski days per year, but they're both solid intermediates for kids. I think Alta has better green and intro blue terrain off the Sunnyside/Albion side, and we've skied off of Sugarloaf and Supreme. My son especially loves the easy glades down above Alf's restaurant.
We have skied Snowbird with them a lot less, never skied Snowbasin. To be fair - we've never done the Baby Thunder area, and I thought Emmas was too much of a choke point for 'Bird's green and blue stuff.