Here is my view as a golfer in the US. In High School, I was a scratch golfer under the USGA HC system. At my home course, I was able to break par on a consistent basis, but I played roughly 2-3 strokes over at other courses on which I had 2 or less rounds of experience (even on courses with 3 or more rounds). It all goes back to knowing the course that you play. I have been out of High school for 4 years now, and only get to play roughly 10-15 times a year back home(if I'm lucky). Hardly enough to remain consistent, but I have been able to maintain a 9. Many people in the US only use their best 10 differentials out of their last 20 rounds... Since I play so little at the current time, I use my last 10 rounds to make it more accurate. After playing some rounds in the UK I have noticed some differences in the playability of the courses and the players themselves.
1. In the US, your better players will have a high ball flight with emphasis on the workability of the ball and spin around the greens whereas the better players in the UK (in my experience) have a low ball flight and let the ball run. In the UK, the courses tend to run a little more than they do in the US and the bump and run is more essential than it is in the US. The wind in the UK is detrimental to a high maneuvering shot, whereas the softer conditions and lack of wind hurt the player with a low runner.
2. The greens in the US have more grain to them so a putt that looks flat, might actually turn the width of 5 or 6 balls or a putt that looks like it will turn may actually go straight. From what I have noticed, the greens in the UK are quicker (due to a lack of grain) than they are in the US. This makes it difficult for American golfers to putt in the UK because they are trained to look for certain factors that are not as prevalent in the UK. Likewise, golfers from the UK that have come to the US tend to struggle on the greens as they get used to accounting for the new factors.
3. Distances. Any top tier golfer knows the exact yardage that their club will fly when hit with their normal swing (i.e. 7 iron goes 167 yards with a comfortable swing). Distances in the US will be different than those in the UK. I found that my tee shot flies 20-30 yards shorter in the UK than in the US (I am from South Carolina so the temperatures are much warmer) and my irons were roughly 5-10 yards shorter. This is critical when looking at approach shots.
As a US golfer, I have been playing roughly 4-5 shots worse while here in the UK. That can be seen as the US having an easier system and one that is not representative of your "real" handicap. That being said, I have only played a few rounds and am still trying to get used to the rollout on approach shots and the distance with the driver. I have hit shots here that would have led to easy birdies back home just to regret it while the shot is in the air because I forgot to take into account the rollout. Next thing I know, I am chipping back to the green and having a 5 + footer for par (rollout again) instead. That being said, I have seen the same problem with British golfers coming to the US and having trouble adjusting to the different conditions. I have only been here (UK) for a month and I am getting better as the time goes on but still trying to lower my ball flight to line myself up with the conditions here. As far as I can tell, the stableford scoring system is the same as what we use back home in our dog fights. I don't know of anybody back home that is happy with scoring in the 20's in an 18 hole dog fight. According the the scoring happiness scale that has been posted on this thread I would say we are the same. The US handicap system is very representative of what you will do in the states just as the system you use here is good at predicting your scoring here. Neither one would be accurate in the other system as players from both places would generally struggle adjusting to the different game. Once you have adjusted to the conditions, however, I would say that your handicap would be pretty interchangeable as you flew across the pond. The courses here aren't any harder than they are back home and vice versa. It's just a different game altogether.
I know I will probably get blasted for this analysis, it is just what I have noticed in my short time here and with my experiences with golfers coming to the US and playing.