In the US, domestic flight First Class is similar to Business Class on European flights, except I don't know of any that have champagne.
Transatlantic, I've flown BA and American. As usual with American, not impressed: the CC are slightly more attentive than in economy, the champagne is there, but you can have problems getting it - seems there's a limited amount. The food is so-so.
BA is very different. In 2008, Mrs radeng and I had a holiday in Australia and New Zealand, and I splurged a load of miles to First on BA. As Mrs radeng settled into her seat on the way out, she said 'I could get used to flying like this!' Result was that after getting on for 24 hours of flying, arriving in Sydney about 0600, we were refreshed enough to spend the day sightseeing. Couldn't have done that flying economy!
Also went on holiday to the US in 2010 using miles to go First on BA. Again, superb.
It's the general ambience, the greater space, the better food, the attention. But you gets what you pay for, although I don't feel that you do with the US carriers on long haul.