The economies of the United States and Japan are generally fairly similar. This is because they are both mixed economies that tend to be fairly market-oriented. However, the Japanese economy is different from that of the US in a number of ways.
Both Japan and the US have basically capitalist economies. Both are based on people being able to own property and use it as they like. However, the Japanese economy is influenced more by its government than the US economy is. The Japanese government for example, has often helped to guide the national economy by deciding what industries should be emphasized. There are more laws governing hiring and firing than there are in the US. The US is, in these ways, more of a free market economy than Japan is.
Another difference between the two economies is in the nature of the firms that dominate. The US economy is more influenced by relatively small firms. By contrast, the Japanese economy is very much dominated by large firms. The US economy is much more open. Smaller firms have more of a chance to grow and become important. It is also worth mentioning that the US economy makes much more use of women than does the Japanese economy.
In general, then, both of these economies are essentially market economies. However, the Japanese economy is more influenced by its government and is also somewhat more traditional and closed than the American economy is.