A^2 is just one matrix multiplication, A*A, which is much faster to do directly than the call to EIG.
For larger n, A^n isn't just doing the multiplication A*A*...*A. You can see the algorithm used by reading the mpower function, edit mpower.
To show a case where EIG is faster, you could increase n significantly, explcitly compute A*A*...*A instead of calling A^n, and have EIG return a vector of eigenvalues instead a dense diagonal matrix D. I think that should work to show an example where EIG is faster.
Note that there are numerical issues with using EIG, in cases where the matrix of eigenvectors P is close to singular. For example, try the EIG method on the matrix A = [3 1; 0 3]:
>> [P, D]=eig(A); product_2 = P * (D^2)/(P)