I have written a couple of posts that tell of some of the differences between schooling in the United States and in Japan (can be found here and here if you haven't read them). Schooling is a major thing so it is relatively easy to find more to talk about.
The Japanese schools are all standardized by the government. Everyone in every school (baring taking a vocational) gets the same education, same or extremely similar books, same teaching methods, and same resources (though some schools will have extra). In the United States this is not true. A underfunded public school in the middle of a major city will give, in general, a much worse education than a sufficiently (or even over funded) private school in the the suburbs. Beyond that in the United States what is taught differs between states. Each state has its own curriculum, meaning that people from different states may learn different things. The US does have standardized testing in the form of the ACT and SAT but the opportunities and education varies a lot for the individual where as everyone in Japan has pretty much the same.
The Japanese go to school an average of 60 days more a year than US students (though some of that time is spent on cultural days, field trips, and sports days). Which means the Japanese receive about a year more of schooling than their US counterparts even though they spend the same number of years in school. Since the Japanese do not have as long summer breaks less information is lost during the breaks. There are several papers/studies that show information is lost between school years. Here is a link to one(technically 39): The Effects of Summer Vacation on Achievement Test Scores: A Narrative and Meta-Analytic Review.
The way the Japanese and the US spend money on education is different. In the United States most public schools are free (paid for by tax dollars) and 40% or more of the money that goes to the school is not spent on education. Instead it is spent on secondary things like transportation for students to school, DARE programs, lunch, and sports. In Japan, public high school is not free, parents have to pay. Japanese schools rarely pay for student transportation most students walk or ride their bikes to school. Japanese schools cut costs by having students clean the schools, i.e. don't have to hire janitors. A lot of the high schools in Japan do not even have lunch rooms and if they do you will have to pay. The Japanese also have fund raising to support the after school activities and school activities, such as cultural festivals when the schools open themselves to the neighborhood and try to sell things (drinks, or charge for a haunted house, or sell home made knickknacks) to help fund the school.
The Japanese curriculum and teaching methods differ from the US teaching methods. Japan has the highest literacy rate in the world with around 99% of adults being able to read. In arithmetic the Japanese high school student is around 2-3 years ahead of their United States counterpart (of course this does matter on state, raw ability, and financial status). I was considering writing about the Japanese teaching method for math and the mathematical curriculum but there is no point someone already wrote a very good article on that, here's a link: Mathematic Education in Japan and the United States Curriculum. For some things like English and History the Japanese aren't that great. The history books are edited somewhat (for example: they deny Japanese actions in China and Korea during World War 2. Of course the US history books often aren't very good as well many being out of date and they are also edited/changed for political reasons). There is a lot of controversy on Japanese History books that many sites talk about, here is a link to a wikipedia article on it: Japanese History textbook controversy wikipedia. While the Japanese do teach English it more concerned with the form than the actual spoken language, which means if a Japanese student came to the US and tried to speak English most Americans wouldn't have a clue as to what they are saying. Here is one article on how English is taught in Japan: How English is Taught in Japan. The Japanese also rely too much on rote memorization of facts. Which is fine for some things (multiplication tables, periodic table of elements) but it is an ineffective tool in mastering a complex subject like the English language.