purposely, purposefully
Purposely is the older word (15c) and means ‘on purpose, intentionally’

• (We've purposely changed our programmes on Tuesday to fit in with the Royal timetable —BBC Press Release, 2002 [OEC])

whereas the more recent word purposefully (19c) corresponds to the adjective purposeful and means ‘with a strong purpose, resolutely’:

• He threw down a handful of sovereigns without even looking at Sam, then strode purposefully after the Squire —S. Langley, 2006.

A third word, purposively (20c), means ‘for a particular purpose’, and is more usual in technical contexts:

• Socialism…would have to be built by active human beings working purposively and creatively —J. Dignan et al., 1992.

Occasionally purposively is used when purposefully is probably meant:

• ‘Gerrart-of-it!’ said the larger of the two, moving purposively towards me —Will Self, 1993.

Modern English usage. 2014.