Καινός is new in the sense of not having existed before, in contrast with old preceding it; νέος is new, fresh, young (which καινός never means), in contrast with subsequent prolonged existence by which a person or thing becomes old. What is old, was once (unless external) νέον; if it disappears and another thing takes its place, this is καινόν.
Thus in Matt. 9:17 (twice); Mark 2:22 (thrice); Luke 5:37-39 (four times); wine is called νέος, that is, fresh, not yet old. Again, in Luke 15:12, 13; 22:26; Acts 5:6 Tim. 5:1, 2, 11, 14; Titus 2:4, 6 Peter 5:5; various forms of the word are used in the sense of " young " or " younger." In 1 Cor. 5 it is " a fresh lump." In Heb. 12:24 it should be, not the, but " a new covenant," διαθήκης νέας. It is a fresh covenant, and just beginning, it is one yet to go on and become developed. It is not here in contrast with the old, which is exactly the point in Heb. 8:8, 13, where the new covenant is designated καινή (as in every other such mention of the New Testament), and the apostle reasons, " in that he saith, A new [covenant], he hath made the first old." That is, here it is new, as contrasted with the old. The attentive reader may remark that this determines the force also of Matt. 26:29, and Mark 14:25, " until that day when I drink it new." It is not here, wine not yet grown old, which would be νέος, but wine after a new sort or of a new kind. It is not the old wine at all. Whereas, in all ordinary cases, οἶνος νέος must not be put into old skins, but into new, ἀσκούς καινούς, skins which had never been used before. So καινός is used of a new tomb, piece of cloth, a garment, report, commandment, doctrine, song, name, city, earth, heaven, creation, all things. Hence in Eph. 4:22-24 they were to be renewed, ἀνανεοΰσθαι, in the spirit of their mind. A young man was to be produced, which was to grow on and up in them. But in fact, as to the καινόν ἄνθρωπον, which they were to put on, it had in no sense existed before. It was, according to God, created in righteousness and holiness of truth. In Col. 3:9, 10, they had put off the old man with his deeds, and put on the new, τον νέον, a young life and conduct which is to go on. Still, in fact, though viewed here as the young man, it is by the power of God a newly formed thing (τὸν ἀνακαινούμενον), renewed into knowledge after the image of Him who created it. That is, though it be a young and in that sense a new life in which we live, it is not a modification of the old. The renewal is the production of a new thing in which we are formed according to the image of God. This is quite in keeping with the two epistles: Ephesians maintains the contrast of a new creation with the old; Colossians, the practical new life in which we live, though care is taken to show that this is an entirely new thing, formed of God, while in Ephesians we find the wholly new man is a young fresh path of life, as regards the practical spirit of our mind.
ΠΑΛΑΙΟΣ AND ἈΡΧΑΙΟΣ, AND RELATION TO ΝΕΟΣ AND ΚΑΙΝΟΣ
Παλαιός—more " the former," ἀρχαῖος—" ancient, antique." You could not say ἀρχαῖος ἄνθρωπος in the sense of παλαιός. Αρχαῖος is opposed to νέος but cannot be so absolutely to καινός. But ἀρχαίος can be neither νέος nor καινός. It may be opposed to both: so may be παλαιός. It is contrasted with καινός, but it is not the νέος-what now begins. In 2 Cor. 5:17 ἀρχαῖα are things which have been of old, a long time; we have a new system or creation. So in Matt. 13:52: they are things καινὰ καὶ παλαιά, the old scribe knowledge, and other new things.