Purposely means on purpose — deliberately rather than by accident. Purposefully means with evident determination and direction. She purposely left early (she meant to); she walked purposefully to the door (with resolve). One is about intention, the other about manner.
The core difference
Both come from purpose, but describe different things.
- purposely — deliberately: He purposely ignored the question.
- purposefully — with determination and direction: She strode purposefully across the room.
The test
Substitute deliberately and then with determination. Only one will fit. "He deliberately ignored the question" works; "he ignored it with determination" does not.
Why purposefully is over-used
It is the longer word, and longer words attract writers wanting emphasis. But it makes a claim about manner and resolve that purposely does not — so using it where you mean "on purpose" describes something you probably did not observe.