Charles Hodge, in his commentary on Ephesians gives a very good explanation of the difference in the quotation.
Hodge reasons that Paul applies the receiving of gifts in accord with the original idea of a King with plunder. A kind has plunder to give to his kingdom, so he switches the verb while retaining the same image and idea:
The divine writers of the New Testament, filled with the same Spirit, which moved the ancient prophets, are not tied to the mere form, but frequently give the general sense of the passages which they quote. A conqueror always distributes the spoils he takes. He receives to give. And, therefore, in depicting the Messiah as a conqueror, it is perfectly immaterial whether it is said, He received gifts, or, He gave gifts. The sense is the same. He is a conqueror laden with spoils, and able to enrich his followers. (Hodge, Ephesians, p217)
Hodge also gives a very good sense of how Paul applies this Psalm to Messiah:
The identity of the Logos or Son manifested in the flesh under the new dispensation with the manifested Jehovah of the old economy. Hence what is said of the one, is properly assumed to be said of the other. Therefore, as Moses says Jehovah led his people through the wilderness, Paul says Christ led them. 1 Cor. 10:4. As Isaiah saw the glory of Jehovah in the temple, John says he saw the glory of Christ. John 12:41. As it is written in the prophets, “As I live, saith Jehovah, every knee shall bow to me, and every tongue shall confess to God,” Is. 45:23, Paul says, this proves that we must all stand before the judgment seat of Christ. Rom. 14:10, 11. What in Ps. 102:25, &c., is said of God as creator, and as eternal and immutable, is in Hebrews 1:10, applied to Christ. On the same principle what is said in Ps. 68:18, of Jehovah as ascending to heaven and leading captivity captive, is here said to refer to Christ.
Therefore Paul is saying as Jehovah came down to rescue Israel and ascended on his throne under King David, taking spoils of other nations and establishing his kingdom, so Messiah came down to earth, redeemed a people ascending into heaven and gave gifts to his church during pentecost.
Interestingly Alfred Edersheim, the Jewish historian, found a reference where this Psalm was in some aspects taken as Messianic:
Ps. 68:31 (32 in the Hebrew). On the words ‘Princes shall come out of Egypt,’ there is a very remarkable comment in the Talmud (Pes. 118 b) q and in Shemoth. R. on Ex. 26:15, &c. (ed. Warsh. p. 50 b), in which we are told that in the latter days all nations would bring gifts to the King Messiah, beginning with Egypt. ‘And lest it be thought that He (Messiah) would not accept it from them, the Holy One says to the Messiah: Accept from them hospitable entertainment,’ or it might be rendered, ‘Accept it from them; they have given hospitable entertainment to My son.’ (Alfred Edersheim Life and Times of Jesus, Appendix 9)