Lexicon

G3338 - metamellomai

Strong's: G3338 · BLB lookup Pronunciation: met-am-el'-lom-ahee Part of speech: verb, middle / passive in form Root: meta (after, with a change) + melō (G3199, "to be a care to"). Literally "to have it matter to one afterward", to care differently about something once it is done NT occurrences: 5 verses in the Strong's-tagged KJV, with 8 tags. The KJV word carrying the tag is "repent" 3 times, "repented" 2, with single tags on "himself", "do" and "will" where the KJV spreads the verb over a phrase (measured) Compare: metanoeō (G3340), 32 verses and 36 tags; metanoia (G3341), 24 verses (measured)

Metamellomai is the New Testament's word for regret: the change of feeling that comes after an act, when it now matters to you differently. The English Bible usually renders it "repent", the same word it uses for metanoeō, so the distinction is invisible in translation. It is a rare word, only five verses, but one of them is Judas, and that verse carries most of the popular teaching on the difference between sorrow and repentance.

All five verses

Matthew 21:29. The first son in the parable: "And he answered and said, I will not: but afterward he repented himself, and went."
Matthew 21:32. Jesus to the chief priests and elders: "the publicans and the harlots believed him: and ye, when ye saw it, did not even repent yourselves afterward, that ye might believe him."
Matthew 27:3. "Then Judas, who betrayed him, when he saw that he was condemned, repented himself, and brought back the thirty pieces of silver to the chief priests and elders."
2 Corinthians 7:8. Paul on his severe letter: "For though I made you sorry with my epistle, I do not regret it: though I did regret it (for I see that that epistle made you sorry, though but for a season)." The verb is tagged twice.
Hebrews 7:21. God's oath to the priest-king: "The Lord sware and will not repent himself, Thou art a priest for ever."

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Semantic range

To regret, to feel differently about a past act. Paul's "I do not regret it: though I did regret it" (2 Corinthians 7:8) is the plainest case. Nothing moral is at stake; Paul simply had second thoughts about a hard letter and then, seeing its fruit, did not.
Regret that turns into action. The first son "repented himself, and went" (Matthew 21:29). Here the change of feeling issues in obedience. The leaders are blamed in Matthew 21:32 precisely for not having this regret, the kind that would have led them to believe.
Remorse without return. Judas (Matthew 27:3).
To change one's mind about a decision, negated of God. "Will not repent himself" (Hebrews 7:21): the oath stands and will not be revisited.

The distinction from metanoeō, stated carefully

The usual teaching runs: metamellomai is mere emotion, metanoeō is real change of mind and life. There is a real difference, but the five verses show the slogan needs care.

What is true. Metanoeō and metanoia are the words of the gospel call ("repent ye, and believe in the gospel", Mark 1:15) and are never used of Judas. Metamellomai is never the word for the response the gospel demands. Its root is about how a thing matters to you afterward; the root of metanoeō is about the mind.
What is overstated. Metamellomai is not always empty. In Matthew 21:29 regret leads the son to go and do his father's will, and in Matthew 21:32 Jesus treats this regret as the very thing the leaders should have felt so as to believe. Regret can be the beginning of repentance.
So what separates Judas from Peter is direction, not vocabulary. Judas's regret went back to the chief priests with the money and then to a rope: "and he went away and hanged himself" (Matthew 27:5). Peter's grief went out "and wept bitterly" (Luke 22:62) and was met by the risen Christ. One sorrow turned in on itself; the other turned back to the Lord.

2 Corinthians 7: regret, repentance, and no regret

Paul puts the two word families side by side in one paragraph (2 Corinthians 7:8-11):

7:8, metamellomai twice, of Paul's own second thoughts about the letter.
7:9-10, metanoia once in each verse: "ye sorrowed to repentance" (7:9, KJV).
7:10, "For godly sorrow worketh repentance unto salvation, a repentance which bringeth no regret: but the sorrow of the world worketh death" (2 Corinthians 7:10). "Which bringeth no regret" is ametamelētos (G0278), the negated adjective from the same root, which occurs only here and at Romans 11:29 in the Strong's-tagged KJV (measured).

The paragraph works as a gloss on Judas and Peter. Worldly sorrow ends in death; godly sorrow produces repentance, and that repentance is the one thing nobody ever regrets.

God "will not repent himself"

Hebrews 7:21 quotes Psalm 110:4: "Jehovah hath sworn, and will not repent: Thou art a priest for ever After the order of Melchizedek." The Hebrew verb in the psalm is nacham (H5162), the same verb used when "it repented Jehovah that he had made man" before the Flood (Genesis 6:6). The writer to the Hebrews renders it with metamellomai, and his argument depends on the denial: the Levitical priests were appointed without an oath, but Christ's priesthood rests on an oath God will not revisit (Hebrews 7:20-22; compare Hebrews 6:17-18).

The companion statement is Romans 11:29: "For the gifts and the calling of God are not repented of", using ametamelētos. Both texts belong to the same theme as "God is not a man, that he should lie; neither the son of man, that he should repent" (Numbers 23:19; 1 Samuel 15:29). See Divine Immutability, and H5162 - nacham for the passages where God is said to relent.

Where the word decides a debate

"Judas repented, so he was saved." The Gospel uses the regret verb for Judas, not the repentance verb, and tells us where his regret went. The question is not whether he felt sorrow but whether he turned to God, and the text records the opposite. See Judas Death Contradiction Objection Defeater for the separate charge that Matthew and Acts disagree about how he died.
"Sorrow for sin is the same as repentance." 2 Corinthians 7:10 distinguishes two sorrows by where they lead. Feeling bad is not the gospel response; turning is. See Repentance.
"The Bible says God changes his mind." Hebrews uses this verb to deny exactly that about his sworn purpose in Christ. Where Scripture does say God relents, the word is nacham and the context is his announced response to human turning, not a revision of his oath.

Notable verses

Matthew 27:3, Judas "repented himself"
Matthew 21:29, the son who regretted and went
Matthew 21:32, the leaders who did not
2 Corinthians 7:8-11, regret and repentance in one paragraph
Hebrews 7:21, "will not repent himself", from Psalm 110:4
Romans 11:29, the calling of God "not repented of"
Luke 22:62, Peter "wept bitterly", the contrast case

Patristic note

The Fathers read the Judas verse the same way, as sorrow without return. Their English translations say "repented", since the KJV did.

Chrysostom on Matthew 27:3: the verse is a charge against Judas "not because he repented, but because he did so, late, and slowly, and became self-condemned" (Homilies on Matthew, NPNF1 10, PDF p. 403). And on Hebrews: "Judas also repented, but in an evil way: for he hanged himself" (Homilies on Hebrews, NPNF1 14, PDF p. 409).
Augustine, counselling a community: "imitate not the repentance of Judas the traitor, but the tears of Peter the shepherd" (Letters, NPNF1 1, PDF p. 487).
Origen draws a softer inference: that Judas brought the money back shows "that the instructions of Jesus had been able to produce some feeling of repentance in his mind" (Against Celsus, ANF 4, PDF p. 321). Regret, on this reading, was real; it simply did not go to Christ.