It occurred to me recently that there’s a difference between “on purpose,” “purposely,” and “purposefully.” Although they’re all similar, they imply subtly different levels of enthusiasm for committing an action.
If a hypothetical lady, Louise, were to throw a kitten “on purpose,” she did so with intent. Her intention was to throw a kitten, and she did. She doesn’t have to have any other reason for throwing the kitten.
However, if she were to throw a kitten “purposely,” this would imply that she had some underlying goal to achieve, and throwing the kitten was simply a step toward achieving that goal. There is some implied (potentially malicious) intent behind the purposely executed action.
“Purposefully” is, in a way, the strongest of the three phrases. It implies that as Louise throws the kitten, her reason for doing so is in the front of her mind and it is strong enough to drive her to action (she is literally “full of purpose”). To do anything “purposefully” gives the action a sense immediacy, and urgency that doing something “purposely” or “on purpose” would lack.
I’m inclined to say that “on purpose” and “purposely” are, in essence, interchangeable. After all, if Louise throws the kitten at a man “purposely,” with the implied malicious intent of hitting him with it, he has every grammatical right to turn around and exclaim, “You bitch, you did that on purpose!” So perhaps the two have the same meaning, and are just used in different forms. Still, I would argue that there’s a very subtle difference between the two in common usage.