The importance of coming to a decision concerning the relative authority of parts of the Bible, parts of the New Testament, and between the Gospels; can be seen by comparing the accounts of (apparently) the same event in the Fourth Gospel ('John') and Luke. (These are quoted in full at the end of this post*.)
Now, although these accounts apparently refer to the same incident, they differ in many details - in particular they differ in terms of the identity of the woman - for 'John' the woman is someone well known and loved by Jesus, sister of his beloved friend Lazarus; for Luke she is just identified as an anonymous woman and someone regarded as leading a publicly-ungodly life ('a sinner').
Most importantly, the 'moral' of the story is different in each - For 'John' it references, I believe, some 'lost' ceremonies of a spiritual wedding of Jesus with Mary (who is, presumably, the woman he married in Cana - in an ordinary Jewish wedding) and a foreshadowing of the deal of the incarnate Jesus, and his burial - and linked with Mary (Magdalene) being the first to speak with the resurrected Jesus. And/ or the moral is about the eternal versus the worldly.
(Since 'John' was writing shortly after the ascension of Jesus, all such contextual details will have been well known to his intended audience.)
But for Luke the story is 'about' the infinite forgiveness or atonement of Jesus, and how this means the most for those with 'the most' sin, those whose lives are built-upon the denial of God - emphasising that Jesus (unlike the Pharisees, and those who regarded adherence to The Law as the only path to salvation) came to save sinners (which is everybody, but particularly those who were furthest from The Law and - up to that point - the most vehemently atheist, selfish, self-indulgent etc).
So, how can we makes sense of these apparent discrepancies? So far as I can tell, we need to assume one of four basic possibilities:
1. 'John' is more authoritative, or
2. Luke is more authoritative; or
3. Both are equally authoritative, and are authoritative (both being valid alternative descriptions of the same event and meaning); or that
4. Neither are correct: neither is authentic, both equally wrong (and therefore nothing of this kind ever actually-happened).
If we regard 'John' as authoritative and the account of a recent eyewitness, then we make sense of Luke in terms of him later collecting scattered accounts of Jesus's life and teachings and - under divine inspiration - making the best sense of him that he could. In this account Luke has done something like conflating several stories into one. This Luke's account of the essential teachings and meanings of Jesus's life is correct (because divinely-inspired); but the historical details are sometimes mixed-up. This is - pretty much - what I believe is correct.
If Luke is authoritative, then 'John' - writing much later, and from a faulty memory, or via an unreliable scribe, or a representative of his division of the early church - has made a mistake based on a partial memory, and perhaps the conflation of various Marys with perhaps unnamed others.
The mainstream view is probably a mixture of giving Luke priority, and also using all available scriptural material pretty-equally, trying to triangulate upon the truth. Perhaps the two accounts are partly complementary, and partly selective. This also goes along with ideas of Biblical inerrancy, or 'literalism' or 'fundamentalism' - which generally assumes that the whole Bible, or, at least, the whole New Testament - or at least the Gospels and Paul's Epistles; are equally valid.
(Indeed, in practice - especially among traditionalist creedal Protestants, Paul's Epistles, rather than any of the Gospels, may be given Primary Authority - and the Gospels are interpreted in their light.)
Rejecting both 'John' and Luke in favour of some unknown, perhaps lost, primary text, variously garbled through several independent lines of transmission, is another possibility, in principle.
My point is that each approach represents different assumptions, and leads to different answers.
We therefore need to be clear about our assumptions - and, I would argue, to trace these assumptions back to our primary intuitions - which may be related to larger units of meaning.
For example many mainstream church-going Christians have a intuition of the validity and authority of a particular Christian denomination or church - and they accepts their detailed evaluations from that particular source of authority.
Others, like myself, try to discover more specific intuitions derived from scripture, church teachng and practice, theology... or whatever - including the prime intuition that these specific intuitions are ultimately valid....
John 12: 1: Then Jesus six days before the passover came to Bethany, where Lazarus was, which had been dead, whom he raised from the dead. There they made him a supper; and Martha served: but Lazarus was one of them that sat at the table with him. Then took Mary a pound of ointment of spikenard, very costly, and anointed the feet of Jesus, and wiped his feet with her hair: and the house was filled with the odour of the ointment. Then saith one of his disciples, Judas Iscariot, Simon's son, which should betray him, Why was not this ointment sold for three hundred pence, and given to the poor? This he said, not that he cared for the poor; but because he was a thief, and had the bag, and bare what was put therein. Then said Jesus, Let her alone: against the day of my burying hath she kept this. For the poor always ye have with you; but me ye have not always.
Luke 7:36 And one of the Pharisees desired him that he would eat with him. And he went into the Pharisee's house, and sat down to meat. And, behold, a woman in the city, which was a sinner, when she knew that Jesus sat at meat in the Pharisee's house, brought an alabaster box of ointment, And stood at his feet behind him weeping, and began to wash his feet with tears, and did wipe them with the hairs of her head, and kissed his feet, and anointed them with the ointment. Now when the Pharisee which had bidden him saw it, he spake within himself, saying, This man, if he were a prophet, would have known who and what manner of woman this is that toucheth him: for she is a sinner. And Jesus answering said unto him, Simon, I have somewhat to say unto thee. And he saith, Master, say on. There was a certain creditor which had two debtors: the one owed five hundred pence, and the other fifty. And when they had nothing to pay, he frankly forgave them both. Tell me therefore, which of them will love him most? Simon answered and said, I suppose that he, to whom he forgave most. And he said unto him, Thou hast rightly judged. And he turned to the woman, and said unto Simon, Seest thou this woman? I entered into thine house, thou gavest me no water for my feet: but she hath washed my feet with tears, and wiped them with the hairs of her head. Thou gavest me no kiss: but this woman since the time I came in hath not ceased to kiss my feet. My head with oil thou didst not anoint: but this woman hath anointed my feet with ointment. Wherefore I say unto thee, Her sins, which are many, are forgiven; for she loved much: but to whom little is forgiven, the same loveth little. And he said unto her, Thy sins are forgiven. And they that sat at meat with him began to say within themselves, Who is this that forgiveth sins also? And he said to the woman, Thy faith hath saved thee; go in peace.