If you were to ask me for the biggest difference between the US and Europe (apart from the sizes of the waistlines), I'd have to say the thing that stands out the most is.... SNEAKERS.
When Europeans do wear sneakers it tends to be only when playing sports, or more trendy kinds of sneakers may be worn by younger people - but these tend to be coloured or dark and certainly not suitable for sports.
You'll usually see a European wearing proper shoes, even when not at work, so unless you want to identify yourself immediately as an American - buy some.
Of course, there are some exceptions to this, and unfortunately people are sometimes seen wearing sandals with white or black socks, but luckily this doesn't happen everywhere in Europe.
Best tip - once you arrive just look at people's feet.
And no sneakers at work please - even if you are allowed to wear casual