In US English "art gallery" would usually mean a private gallery where the pieces are for sale, and "art museum" would usually refer to a public space where the pieces are on display for the education and entertainment of the visitors.

Both the previous answers are correct, but they relate only to AmEng. Congratulations to David for being clear about this. In English-speaking countries of other continents, a gallery exhibits art (Tate Gallery, National Gallery of Australia etc..), a museum exhibits objects of interest - history, archaeology, natural history. Update: there are now four more answers all saying exactly the same correct, but limited, thing. I'm afraid they all seem to confirm the view that Americans don't travel much. Apart from General MacArthur of course, who presumably took 'museum' with him...

I think the art museum is better

My guess about the National Art Center of Tokyo was correct. The National Art Center of Tokyo is not a "gallery." It is a museum. However, the National Art Center of Tokyo contains "galleries" within it. Look at the floor map: http://www.nact.jp/english/information/floormap/ On the first floor, you see "Gallery 1A/1B," "Gallery 1C/1D," and "Special Exhibition Gallery 1E." These are not "art galleries," though. They are just "galleries" or "museum galleries."

"Gallery" is spelled with an "e." In the United States, in the context of art, the basic meaning of "gallery" is "a collection that is on display." A gallery is a room or set of rooms where pictures can be seen. The pictures will be related to each other in some way. The phrase "art gallery" suggests a private business where pictures are bought and sold. An "art museum" means an institution devoted to collecting and showing art. Usually it will be a non-profit organization. A museum has "a collection." The collection consists of all of the pictures it owns. It may have more than one collection, devoted to different things. Only some of the pictures in the collection will be on display. A museum has "galleries," usually more than one. Each gallery is a place where a group of related pictures is on display The Boston Museum of Fine Arts has a gallery of paintings by Monet. It has a gallery of ancient Egyptian art. It has a gallery of Art of the Americas. I would not call the National Art Museum of Tokyo "a gallery." I would expect it to _have_ "galleries."