Yes the other specs are important, probably the most important is the watts on the 12v rail or rails because that will decide what Kind of video card you will be able to use.
And also there is a difference between cheap and expensive (name brand) power supplies Not only the amps on the 12v rail but also just the lasting quality. A CHEAP one may damage your system or die off quickly, and expensive ones are usually more efficient meaning they generate less heat, last longer, and use less energy, meaning you can get a lower wattage one.
FOr expample You would be better of to run a nice 380 watt PS of an system that draws 300 watts then a crappy 450