Maximum-clique practical applications
If you ask questions like that, I feel that the only answers you could get are problems that "reduce to the maximum clique". That would be a mistake. There are many problems in practice where finding a maximum clique is thought of as a subroutine, as it is not the most expensive part of the algorithm.
Finding cliques is known to be NP-Hard, and depending on the instances it is or it is not. NP Hard does not mean anything in practice.
My (reduced) own experience tells me that I would prefer 1000 times to deal with a maximum clique problem that with a TSP, or worse : an integer multiflow.  Max clique is for me one of the "kind" NP-hard problems.
And it means that I sometimes reduce problems to Max Clique because it can be done this way, and because I know that I will get the result fast in practice (and also because I already have all the functions available, so there is nothing new to code).

If you are interested in pratical problems that can be formulated immediately as a max clique problem, I am afraid you will not find many. Practical problems have their own aspect. However, I solved one thousand time "Max Clique" problems, often on optimization problems which would not have required it, because Max Clique is, somehow, an easy problem, and because it is so general of aspect.

I guess the last time I computed a maximum independent set (which is exactly the same) was in order to obtain a decomposition of a graph (a partition of its edges into copies of a given graph).

The graph had hundreds of nodes, and solving the max clique problem on it was very far from being the bottleneck. 

Hoping it helped,