As a 19-year-old Western college student whose first true experience "away from home" just came to a close, hopefully I can give you what you're asking for.
For me, making the first transition away from home was quite difficult - I moved into my university's dormitories this past September, and it was quite the transformational experience. Home had been the only place I'd really ever known, and there was a large degree of comfort I associated with it. When I packed my life up and moved out, suddenly I had to deal with a void in my life that I had frankly been unable to prepare for. I can't say my first term of college was horrendous - it wasn't - but I certainly didn't like it, and the whole time I missed home and looked forward to going back. Fortunately enough, my school's schedule meant that I only stayed there for one 10 week term in the fall, after which I had a 6 week break to return home. When I did, however, things changed. I was home, but at the same time I wasn't. Home, for me, had changed fundamentally - it was no longer the place I'd be spending most of my time, and I knew that my place there was only temporary because I was going to have to go back to school. It was at this point that Western culture began to truly make its mark on me, because I had to come to terms with the fact that home could never be what it used to be (since I was expected to come and go but never again stay for good).
I then went off to my next school term in January, and had a much better experience - I was much, much more involved in activities and organizations on campus, I met many more people than I had met in my first term, and things just felt right for a change. "Home" for me was no longer a fixed place, because I now held the power to put "home" wherever I wanted to. Home became the familiar sights and sounds of campus life; it became the faces and voices of friends.
I think that's where the fundamental difference lies - self-empowerment. The notion that I alone have the power to choose where I go in life is absolutely crucial to my functioning effectively in Western society, simply because the power of the individual is so key to Western philosophy in all of its manifestations. The idea of an everchanging home isn't so much a desire to be nomadic as it is a desire to self-actualize and pursue one's own ends, and my personal desire to be independent and achieve big things necessitated my reinterpretation of the concept of "home." However, I know many other Westerners who are content to stay in one place all their lives, so perhaps their own personal desires don't require them to uproot themselves at all.
I don't think a love for the feeling of being "on the road" drives Westerners to move: I think it's the self-confidence that comes with personal responsibility for directing our own lives however we see fit.