Stories in these games are MGS-flavor wacky jappy shit. Great stuff.
I'll play Yakuza 3 eventually but I know there were some situations in Yakuza 1 that carried into Yakuza 2. There are reoccurring characters even in the side-missions.
As for the game sand how much they're different, there isn't much change between them but I'm fine with that. Biggest change between Yakuza 1 and 2 in the core game is much better combat. Yakuza 3 seems to just have the "tech" upgrade and what comes with it (free camera as you run around, better graphics, etc). Judging from the Yakuza 4 demo, the combat and camera is quite a bit better than Yakuza 3 without even taking into account the three new, different playable characters.
I don't mind that they mostly stay the same because it's a guilty pleasure game. A new game is a new story, sub-missions, arcade games, etc. to go through. You mostly go through the same motions each time. Reminds me a bit like the Castlevania GBA/DS games. I enjoyed Aria, DoS, and PoR even though they're basically SOTN. Very unambitious but I was content with playing one every year. It's like comfort gaming. Then OoE came out and sucked hard.