The two words, ἁμαρτία (hamartia = sin) and ὀφείλημα (opheiléma = debt) are literally very different. However, their idiomatic theological use in the NT, when uses metaphorically, are almost the same.
- ἁμαρτία (hamartia) occurs about 173 times. It is defined as (BDAG #1), "a departure from either human or divine standards of uprightness", or, (Thayer #2) "that which is done wrong, committed or resultant sin, an offence, a violation of the divine law in thought or in act".
- ὀφείλημα (opheiléma) occurs only twice: Matt 6:12 used theologically; and, Rom 4:4 used literally. BDAG defines these two words as:
. (1) (literal) "that which is owed in a financial sense, debt"
. (2) (theological) "obligation in a moral sense, debt = sin"
Thus, it is immediately apparent that as used in the Lord's model prayer they two are essentially equivalent. Both signify sin or sinfulness for which the model prayer seeks forgiveness.
It is not known what Jesus actually spoke on this occasion whether in Aramaic or Greek (more likely the former). However, Thayer has a suggestion which is quoted below as part of the entry under ὀφείλημα (opheiléma):
b. in imitation of the Chaldean חוב or חובָא (which denotes both debt
and sin), metaphorically, offence, sin (see ὀφειλέτης, b.); hence,
ἀφιέναι τίνι τά ὀφειλετα αὐτοῦ, to remit the penalty of one's sins, to
forgive them, (Chaldean חובִין שְׁבַק), Matthew 6:12. (Cf. Winer's
Grammar, 30, 32, 33.)