The two sources of applications come from two sources of examples of Alexandrov spaces:

1. Limits of Riemannian manifolds with lower curvature bound.

2. Quotients of Riemannian manifolds by an isometric group action with closed orbits. 

Your example, the finiteness theorem, is of type 1. Another example is the upper bound on integral of scalar curvature of Riemannian manifold in terms of its lower bound on sectional curvature, diameter and dimension, see [my paper][1].
There are many more, in fact (1) provides the main sourse of applications so far.

The examples of the second type include the classification of 4-dimesional Riemannian manifolds with positive/non-negative curvature; it was done by Grove and Wilking [here][2] and based on earlier result of Hsiang and Kleiner.
Yet anther example is the optimal bound for the number of certain type finite subgroups up to conjugacy in a crystallographic group, see [this paper][3] by Lebedeva (which is build on an observation of Perelman, which is build on an combigeometrical problem of Erdős, Danzer and Grünbaum). 


  [1]: http://mi.mathnet.ru/eng/aa508
  [2]: https://arxiv.org/abs/1304.4827
  [3]: https://arxiv.org/abs/1111.7253