Clearly, both of these departments work on problems.
However, it is most commonly accepted that a help desk is inward facing and used by internal staff. Customer Support is outwardly focused and used by people external to the business. Usually customers. (Makes sense)
For further differentiation, here is a list of functions generally found in help desks but not customer support:
Tracking configuration changes
Tracking IT inventory
Managing IT assets
Developing reports for various uses
Meanwhile, the customer support desk can be found doing these where the help desk generally doesn’t:
Up selling or cross selling
Planning marketing activities
Dispatching field service
Implementing personalized customer experiences
Whew! I’m glad we got that out of the way. I hope you find it instructive. If anyone has anything to add, we’d like to hear it.