The real-time aspect, like Rummy mentioned, is by far the best thing about the game. WildWorld was fantastic, because you'd boot it up and something was different or new everytime. I loved going to the bar and being able to collect a new KK song on certain days, was genuinely gutted if I missed out.
It's got that Nintendo charm, although the Wii version hasn't expanded on the formula enough for me. It's hard to compare it to another franchise that Nintendo do, because it isn't designed "for girls" or for anyone above or below a certain age. It really is one of those games where everyone can enjoy it, due to the simplicity of it.
What I would like though is more things to do, plus I would love to expand the village/city, so you could add a ton more things to it. The 3DS version seems to be heading that way, but even then I'm not sure it's enough.