The difference between the words "buono" and "bene" is mainly in their position and so in their role in a sentence. While "buono" is used as an adjective (to describe a person/object), "bene" has the function of an adverb as it describes the way an action is carried on.
Ex. ESSERE (to be) + buono: Il papà è buono. (Dad is good.)
VERBO (other verb) + bene: Parlo bene l'italiano. (I speak Italian well.)