Shenmue combines several genres. It contains elements of action, adventure, mystery, martial arts, "walking simulator", and even racing to a degree. It's a completely unique experience. Very few games make the player feel like they are in a living, breathing world with real people, not just repeating NPCs.
(Contrary to the hate a few posts up) I think Heavy Rain is one of the few games to captures the spirit of Shenmue, albeit in a much smaller and linear scope. Heavy Rain surpasses Shenmue in the sense that the players choices drastically effect the narrative and ending. In Shenmue the player is dragged along a strict story arch. Players can decide how they get from Goal A to Goal B, but ultimately no matter what happens Ryo's going to fight Chai in the Arcade, work at the harbor, and eventually board the boat to Hong Kong. There are minor changes based on player choice (if they beat Chai, for example), but most of them are insignificant to the overall story (Telltale Games Syndrome).
With that being said, Shenmue is still one of my favorite games of all time. There are very few flaws (most can be attributed to it's age). It may not be as refined as Heavy Rain but it is a much more ambitious title. Heavy Rain is an interactive movie, where as Shenmue is an interactive world. Both are good in their own respects, but Shenmue will always be one of gaming's most impressive feats to me.