I've seen a lot of math debate, calculations and other stuff related to Rubik's cube lately, but I don't really understand why is it important, why should anyone spend time asking and answering questions about Rubik's cube, and what good brings Rubik's cube to mathematics?
I have also found something that might be interesting, after a bit of google search.
Numberphile have made a couple of videos about it, and they are quite nice:
Numberphile playlist on Rubik's cube
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Is a good pedagogical example for teaching and illustrating finite groups properties. And is a famous 3D game!. But is not more important than rest of examples in group theory and mathematics.
Rubik cube is isomorphic to the following finite high order group,
$$(\mathbb Z_3^7 \times \mathbb Z_2^{11}) \rtimes \,((A_8 \times A_{12}) \rtimes \mathbb Z_2)$$
Is not more important that others high order groups like monster group http://en.wikipedia.org/wiki/Monster_group, monster group is of greater mathematical interest that the rubik cube since the monster group is a simple group with too many elements.
The fact that the order of an element in a group divides the order of it, can easily illustrate with a rubik cube when you are teaching groups. Take a solved rubik cube and make any combination. For example moving two adjacent sides. Repeating this combination in theory should get to cube armed again since the group representing the rubik cube is finite. (it takes 63 repetitions. Try it!)
Here is a java animation showing a maximal order combination (order 1260)
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The reason why Rubik's cube is important in mathematics is because set of all cube moves in a Rubik's cube form a group. So, it is an interesting example of finite group. You can find more information on this from this paper. Also, study of rubik's cube can lead to some interesting questions and whose solutions can in turn lead to some interesting mathematics.
Often, mathematicians like to tackle a problem because it is either difficult or interesting. In the case of Rubik's cube, the associated mathematics can fall in to both categories. The necessary mathematics is also closely related to very important subjects such as combinatorics, computational complexity, and group theory, to name just three.
Also, the interplay between these subjects can often lead to interesting and sometimes surprising results. It is therefore a worthy cause to solve these problems, as the solutions can often shed light on problems in other areas that are much more applicable to real world situations (the often cited example of the role of number theory and algebraic geometry in cryptography is a good one).