Some benefits off the top of my head (will amend this list after I do some more research:
Almost-sorted sets benefit from being sorted by heapsort.- Space-conscious environments often prefer the O(1) space complexity of heapsort. Think embedded systems.
- Huge data sets benefit from the guaranteed running time of O(nlog n) as opposed to the probable better running time of quicksort. Think medical, space, life-support, etc.