Do 120-width tires really work better on the rear?
The Ninja 250's rear rim (2.5" wide) was designed for a 120 width tire; the EX250E (86-87) model came stock with a Dunlop K530 120/80-16 rear tire. The EX250F (88-07) model has the same size rim, but it came with a 130/80-16 Dunlop K630. Kawasaki actually put a wider tire on their own rim than is ideal for the wheel width; the 130 is slightly pinched in the rim, while the 120 is right at home. In fact, Avon and Dunlop recommend a 2.75" rim for their 120/80-16 tires, and wider ones for size 130/90-16, which is the usual rear replacement.
This is not to say that 130/90-16 tires don't work well on the EX250. Since that is the smallest size made in many decent tire models, that is what gets used.
A 120/80-16 typically weighs four pounds less than a 130/90-16. This means less dead weight on the chassis, less rotating mass for the engine to turn, and less unsprung weight for the suspension to carry. The lower profile is also more rigid. The tire width is close to the ideal size for the 2.5" rear rim, so the contact patch should be better, depending on tire model. Narrower tires also have quicker steering response.
Downsides to using the 120/80 include a loss in final gearing (3% less than a 130/80-16, 6% less than a 130/90-16) and a loss in ride height (about 1/2"-1"). A 120-width tire is a good choice for around-town riding and tight cornering, where the quicker performance can make a difference. For commuting and touring, a 130-width tire will improve high-speed, straight-line stability.