This section has been started only because it is a common question. The information in the routing section shows that direction is basically a personal decision, however the weather may decide travel direction, depending on your duration and route.
Western Europe, Japan, and North America can be the most expensive part of your trip, and will change little compared to the rapid Westernization of Asia. Eastern Europe can be both one of the cheapest and most rapidly-changing parts of the world. Also, consider that if you miss some parts of Europe, they may be the easiest countries to return to, especially when you are older. However, do not base your trip on expenses alone, or you might let many once-in-a-lifetime opportunities pass you by.
"A major factor in deciding which direction you take can be the speed with which you plan to travel. If you are on a quick round-the-world trip, then going westward is important, because you do not have to keep getting up 'early' to catch the next plane. (I once had to go around in a week or so, and going westward made a tremendous difference.)" <Larry Cotter>
"I found travelling west to be much easier as far as jet lag goes. For taking reasonable length flights, such as LA to Hawaii, or NZ to Australia, you end up adding 3 or 4 hours to your day. For most people, this is much less disruptive to sleep patterns than trying to go to sleep 4 hours earlier than usual. More noticeable for the longer flights, of course." <Dave Patton>
"Most European travellers start with Asia, and you may find you get on a bit of a track. We found it refreshing to be going the other way to most people, and it also leads to a more productive exchange of information." <Chris Finlayson>
"I am often asked why I started in Asia instead of Europe. The main reason was to arrive in Asia in good spirits and health, with plenty of cash. I felt that if I saw enough of Asia but had to leave due to unforseen circumstances (health, family, etc), then at least I would have met my goal. Asia helped me learn the most about myself and my culture by its sheer difference, which Europe could never do. I found Europe to be extremely expensive, and Asia had taught me to not spend good money in Europe that could last me 5-10 times longer in Asia. I was bored silly after my second day in Europe. I missed Asia terribly and had to drown my sorrows in Belgian chocolate." <Marc>