The adverbs purposefully and purposely have very similar meanings, but the difference is important. They are derived from the noun purpose meaning “the reason for, or goal of, an action”—so these words both describe acting with reasons and goals.
Purposefully describes acting with a purpose, or like one has a purpose:
Diplomats use language most purposefully; every word is chosen with purpose.
Caroline walked purposefully, so that nobody would stop her.
Purposely means “not accidentally”—acting with intention, not randomly. We can also say on purpose, or intentionally:
Did you call me purposely, or was it an accident?
Jason was purposely not invited to the party, because he drinks too much.