Image via WikipediaPurposely: adverb - on purpose or by design; done with intention, deliberately.
Purposefully: adverb - having a purpose or meaning; designed or intentional; with a definite purpose in mind; resolute.
What you do 'purposely', you do deliberately.
What you do 'purposefully', you do resolutely and with some purpose in mind.
'Jason placed the bar of wet soap purposely to see Jane step on it, slip and land in the bath full of cold water.'
'Jane purposefully surveyed the floor before entering the bathroom, knowing Jason was likely to try to play some dirty trick on her.'
15 September 1971 Greenpeace was founded. I joined them later in the same decade, about 1976, if I remember rightly.