When it comes to wine, yes indeed the old is better.
The context for the above verse, however, is not about wine, but rather about the Old and New Covenants:
"And he spake also a parable unto them; No man putteth a piece of a new garment upon an old; if otherwise, then both the new maketh a rent, and the piece that was taken out of the new agreeth not with the old. 37And no man putteth new wine into old bottles; else the new wine will burst the bottles, and be spilled, and the bottles shall perish. 38But new wine must be put into new bottles; and both are preserved. 39No man also having drunk old wine straightway desireth new: for he saith, The old is better." (John 5: 36-38)
There is no mixing the old and the new.
Either we must live under the Old Covenant completely, which ministers death and a complete need for His grace and life, or we accept the New Covenant, in which He provides all things for us.
When it comes to the covenants, or the ordering of God's world and His relationship with us, the Old kills, but the New grants us His life and blessings:
"“And such trust have we through Christ to God-ward: 5Not that we are sufficient of ourselves to think any thing as of ourselves; but our sufficiency is of God; 6Who also hath made us able ministers of the new testament; not of the letter, but of the spirit: for the letter killeth, but the spirit giveth life.
“7But if the ministration of death, written and engraven in stones, was glorious, so that the children of Israel could not stedfastly behold the face of Moses for the glory of his countenance; which glory was to be done away: 8How shall not the ministration of the spirit be rather glorious?" (2 Corinthiams 3: 4-8)
"For if that first covenant had been faultless, then should no place have been sought for the second.
"8For finding fault with them, he saith, Behold, the days come, saith the Lord, when I will make a new covenant with the house of Israel and with the house of Judah:" (Hebrews 8: 7-8)
Notice that God found fault with the Old Covenant and has ushered in the New through His Son Jesus (Hebrews 9: 15).
In spite of the greater glory of the New Covenant, Jesus teaches that going from the Old to the New is not easy.
Adam and Eve were tempted to know good and evil for themselves. The Israelites wanted to defeat the giants on their own. Even when God gave them the land, they refused to rest. Martha preferred to serve rather than be served, and yet Martha's sister Mary had the better part (Luke 10: 42)
The New Covenant is an acquired taste for us, as we grow more in His grace and less in our efforts.
To rest is real labor for us, and yet were called to enter into this rest, where we believe in Jesus and his great caring for us (Hebrews 4:11)