There is a difference between “purposely” and “purposefully.

“Purposely” is “intentionally” or “on purpose,” or perhaps, “deliberately.”  It is the opposite of “accidentally,” “by accident,” or “by mistake.”  Most of the time, “purposely” is the word you want.  Like, “Betty purposely used the incorrect word,” or, “He purposely left the door unlatched.”

“Purposefully,” while used increasingly frequently, is seldom used correctly.  Indeed, most of the time speakers use “purposefully,” they mean “purposely.”

“Purposefully” means “with determination,” or “resolutely,” or, “with a (particular) purpose in mind.”  Like, “He walked into the room purposefully, with fire in his eyes.”

I suspect that this is just another case of some people thinking that using one word, here “purposefully,” will make them sound smarter than using another, here “purposely.”  It is unfortunate that, while trying to sound erudite, they just sound ignorant.