I don't think the shopping 'experience' will alter that much, but there will be differences in availability, delivery times, shipping costs, and so on.
That's not to say one region is always getting the thin end of the wedge. The wins and losses seems to see-saw, so it's not like they have a favourite customer base - all other things being equal.
Lego doesn't seem to employ much consistency even within each region, nevermind globally. So it might take getting used to, but is essentially feels like the same store if you can resist comparing the differences obsessively.
The 'Help' section on the US/EU site will contain most of the key information for you to compare.