These examples may help you to get a feeling for the tiny difference between bestimmter and besonderer:
Er trinkt gerne Wein, aber nur bestimmten.
Meaning: he likes drinking wine, but only a certain type (sort / year / from a certain producer)
Er trinkt gerne Wein, aber nur besonderen.
Meaning: he likes drinking wine, but he wants the wine to be something special, something unusual; on the other hand, he is not fixated on one specific sort of wine (as in bestimmter Wein).
An bestimmten Tagen gehe ich essen.
I am used to eating out on certain days. This might be every Tuesday and Thursday. Or every 12th of a month, or whatever.
An besonderen Tagen gehe ich essen.
I am used to eating out on special days / days that are not average days but provide some specific reason for doing something extraordinary, e.g. a birthday or so.
Note: selecting from these two words is not a question of "correct or wrong". It is a question of "what do I want to express".