The Bible uses wine in a wide variety of ways. It can represent a drug or a blessing. It can be a symbol of debauchery or of abundance. Wine was part of the drink offering, symbolizing being poured out in service. It was part of Melchizedek's blessing on Abraham, and 2,000 years later, Jesus uses it in the Passover as the blood of the New Covenant. Psalm 75 shows a cup of wine of God's wrath, and Revelation 18 depicts a cup in the hand of Mystery Babylon, representing its intoxicating culture and the spirit of the times.
Obviously, not all of those meanings are in view here, but when we link the new wine with Jesus being "taken away" (verse 35), it coincides with the Passover cup, representing Christ's blood and the New Covenant. When we add the fact that the Holy Spirit could not be given until Jesus had gone away, then the new wine entails more than just forgiveness, but also suggests God's Spirit—His love, power, and sound-mindedness (II Timothy 1:7).
In the example, the new wine is expansive. The fermentation process produces a great deal of pressure. An old and brittle wineskin will not be able to withstand the increasing stress, and it will burst.
The wineskin is a type of vessel. Throughout Scripture, vessels are symbols for people. For Christians, there is an “old man” and a “new man.” The old man represents the life we had before conversion, and the new man, the new vessel, is the life that comes because of conversion. But if we take the expansive and dynamic new wine, and we attempt to put that into the old life, we can be sure that we will have a disaster on our hands.
Our old lives, our old ways, are entirely incompatible with the new wine. The new wine requires change, expansion, and steady improvement, while in the old life, there was no real desire or ability to change. Remember, the new wine is tied to the blood of Passover, the New Covenant, the receipt of God's Spirit, and the spiritual result that will be produced by those powerful factors. Trying to cram all that into a person who is unwilling to change will invariably result in his coming apart at the seams. The precious new wine is spilled on the ground and dreadfully wasted.