Other advantages in addition to what was already stated, lower power draw: there's no platters to spin up, cooler operating temperatures: again no moving parts to generate all that heat, as a result, usually longer operating life before failure.
As for gaming, it will almost certainly improve your game performance. Most games rely on scratch disks or multiple retrieves from data files off the hard drive throughout the course of your game. Games are so large it really is impossible to load everything it needs into RAM. Games that are level based will obviously have much quicker loads when the level starts, but if you play seamless gaming world type games, having an SSD will cut down on the stuttering when going to new zones and the like.
Also, don't think of SSDs as the holy grail of gaming, yes they'll give you a performance boost, but only if you have the CPU, RAM, and Gfx to support it. Otherwise you're just putting a spoiler and rims on a Pinto.