The game can be played solo, as well - and must be in order to gain access to later areas - but online play yields better items, which is the prime incentive of play. The game's scenario being relatively quickly completed, one then is supposed to beat the game again on more difficult settings, which will be rewarded with better items.
PSO was reportedly going to be released for GameCube, after Sega abandoned its hardware. However, as of this edit, it is still MIA. Sega said that GameCube and DreamCast? players would not be able to play together. The GameCube version has been released in Japan. We're still waiting in the US, though it's supposed to come out sometime in the Fall.
The 'v2' release is less of a sequel than an upgrade. It added soccer games to the 'lobby,' a new area, a new difficulty level, and bumped up the maximum level of characters from 100 to 200. I think they added two new character classes. However, they also changed the payment plan - PSO (v1) was free to play online with the key you got with your game, whereas PSO (v2) required a monthly fee. Sadly, with Sega discontinuing its SegaNet? service, PSO v1 players may be left out in the cold shortly. It's unclear (to me) whether PSO v2 service will continue.