As internpete explained, it depends on what you want to do and see! Golden week is certainly much more expensive, and accommodation is tough to find.
If you want to ski or snowboard, then the winter months are certainly the best time to travel. The north eastern mountains on the Honshu island and the mountains in Hokkaido are covered in thick powder snow in winter.
For picturesque hiking and sightseeing - fall or spring is best for the majority of Japan. Seeing mountains covered in autumn colours, or sakura lined streets and parks is beautiful.
Hiking or tour-cycling around Hokkaido is best to do in late spring, summer, early fall - it's not as hot (usually) as in the more southern parts of Japan.
If you are travelling during the later summer months through Okinawa, Kyushu and Honshu, be prepared for *hot* weather, lots of rain and humidity - it's the wet season. The wet season makes the waterfalls spectacular.
For onsen hopping, I prefer the cooler - colder months. Cooking like a lobster in hot pools during hot, humid weather is not my idea of comfort!
There are many cultural festivals all over Japan - perhaps you can plan a trip around those that interest you, get a JR Pass and take the shinkansen between destinations.