Sakurai has stated that the 3DS will have a more expansive single-player component, and that the design philosophy is that the 3DS version is like a personal training ground, and that the Wii U version is the big stadium where all of the magic happens.
I'm willing to bet that the Adventure Mode in the Wii U version (if there is one) will not be as expansive, and may be a quicker deal, possibly like the Melee one. However, I think the 3DS one will be like Subspace minus the cutscenes (it has to be, because if you're on the bus or whatever, you're likely not going to be able to use the online mode). I also think the Wii U version will have a more expansive online mode, such as a tourney mode for online. 3DS online will be more for just hooking up with friends. There's also the customisation that is exclusive to 3DS. And the stages and trophies, of course. I have a feeling they may implement Street Pass Challenges and Play Coins too.
It really feels like there are designing both versions to encourage you to get the other version for a more complete experience, especially since they both release on separate dates. Which is fine by me.