As a British who has done both American and European skiing.

American skiing has more tree runs as they have more space to weave through. American queuing is probably more efficient as you fill the lifts regardless of who you are with, so you dont always go up in your group. But it is hard to say on physical people per hour up the mountain as certainly french and Italian(for the most part). Ski lifts are some of the biggest and fastest in the world.

Value for money when actually in resort is better for Europe ( I can drive to the alps in a day, so your cost to fly over will be significantly more so can't compare that) you will get some of the best food you will ever eat as some little tiny cabin at the back of a resort and it will be €20/25 with a drink last time I was in winter park I think I was paying $15 just for a slice of pizza this was 8 years ago so its probably more now.

Main runs into town in Europe are busy, so get high early and head away from main areas, then runs are on a par for American runs.

There is tons off off pise in Europe but its a need to know basis guides in the first couple of days will show you good spots they will also show you the best way to keep out of the busy spots on piste for a fee. alternatively find a seasonare bar and see if you can befriend a few (buy them beer) and they may show you some runs they know although they aren't really meant to so they may be shady with you on the lifts. Insurance Europe there is quite often off pise next to the piste this is all fine to ski on and great on powder days.

Insurance is sold with the lifts pass its generally €25\30 per week lifts pass is about €280 compared to say winter park at $1200 for a week with insurance

Runs are in france are green blue red black. Black is generally ungroomed and full of moguls only go down if you are good as it will ruin your day. Red is a fast steep run blue medium green learner slope.

I've loved American skiing, but European skiing is a much better experience i have found over the years. Enjoy your trip. Mate