It's usually a combination of different factors, with the exact combination varying from person to person. Some advantages of Japanese citizenship include:
•The right to reside and return to Japan, no matter what. Even Permanent Residency status can be lost, but citizenship is forever.
•Your children will also be citizens, regardless of the status of their other parent.
•The right to vote, and to run for public office.
•A Japanese passport. While most developed countries' passports are overall equally useful for getting into other countries around the world, they can and do very alot when looking at any individual country (relevant examples here include Cuba, India, and China).
•Japan will fight for you. Hopefully you'll never need it, but the Japanese government really does go to work in getting its citizens abroad home safely when they wind up in bad situations.
•If you're American, you no longer have to worry about double taxation, nor will you have to report the exact contents of all your bank accounts to the US government.
On top of these objective advantages, there are less tactile reasons a person might want citizenship, such as feeling like you're making a real commitment to Japan, a show of loyalty that says you'll be here come hell or high water, or a desire to no longer be counted as a resident foreigner.
Or maybe you just want your legal name to be in Japanese script. ;)
P.S. - There's actually even more advantages listed here:
http://www.turning-japanese.info/2012/06/why-would-anybody-want-to-bec...
