There tends to be a calm calculation behind determined. Roll the sleeves up and bear down on the problem.
A focused energy into a strategic plan where head and heart work together towards achieving the objective.
Desperate on the other hand is back against the wall stuff- emotional, fight or flight in a crisis. There can be power in desperation.
Sometimes situations require a determined response, sometimes desperate.
But desperate is not sustainable over the longer-term. Go to that well too often and you'll find it dry.