Beautiful and simple. There's just one problem; there's no way for O to ever win. X can take the center cube and no matter what O does, X will connect 3 unless X makes some horrible mistake. That's why tic tac toe works in a 2d board. Even Connect 4 was developed with this in mind.
The only ways I can think of to make this fair is to add a third player or to require that the game has to be won on one of the faces and disallow the center cube as an option.
Otherwise, nicely done. The interface is simple and easy to navigate, and it is pleasing to the eye.