"Director" will always be a radar controller. "Approach" will usually be a procedural controller. He'll be the guy who steps you down in the stack, maybe reads out the weather, etc. The Director will be the one who vectors you towards the runway/ landing aid.
"Approach" is the term for the Approach Control facility at an airfield but more and more are being integrated with Terminal Area services so the old "Approach" facility no longer exists. Most large "Approach" facilities are 100% radar nowadays so the controllers will all use the callsign "Director".
At smaller airfields the Approach facility still exists as part of the aerodrome ATC facility and in many cases there is no radar so you'll only get "xyz Approach".