The key to this passage is found in Matthew 9:15, Mark 2:19-20, and Luke 5:34-35:
Can ye make the children of the bridechamber fast, while the bridegroom is with them?
But the days will come, when the bridegroom shall be taken away from them, and then shall they fast in those days.
John the Baptist's disciples and the disciples of the Pharisees did not have the bridegroom. John came to prepare the way for the Messiah and those disciples of his who did not recognize Jesus through John's teaching and change allegiance (as did Andrew in [John 1:35-41]) were still drinking "old wine". The disciples of the Pharisees were in an even worse position as they were being actively taught to reject Jesus as the Christ and they were drinking "old wine" as well.
Although fasting (in particular) is in view here, the 'old wine' versus 'new wine' dichotomy is describing the totality of religious activity under two covenants. Since fasting is strongly related to mourning and, since this mourning is ultimately sourced in man's separation from God, it is not surprising that Jesus disciples could not fast in His presence ... God was with them! A time was coming when His immediate presence would be removed and mourning and fasting would begin again for a time. Ultimately, though, mourning and fasting will cease when Jesus returns.
Those who have become accustomed to old covenant activity (old wine) are not immediately interested in abandoning that old covenant in favor of a new covenant (new wine). Notice how Luke says, "No man also having drunk old wine straightway desireth new". Often, a mix of the two is sought in order to retain the comfort of the old and access the blessing of the new. Jesus says that this practice ruins both:
And no man putteth new wine into old bottles; else the new wine will burst the bottles, and be spilled, and the bottles shall perish. But new wine must be put into new bottles; and both are preserved. - Luke 5:37-38
The apostle Paul has a lot to say about those who intend and attempt to bring Jesus' disciples back under the the old covenant, those who want to put the new wine of the Spirit into the old wineskin of Law:
For there are many unruly and vain talkers and deceivers, specially they of the circumcision: Whose mouths must be stopped, who subvert whole houses, teaching things which they ought not, for filthy lucre's sake. - Titus 1:10-11
And he sternly warns the Church not to leave the Gospel and return to the Law:
O foolish Galatians, who hath bewitched you, that ye should not obey the truth, before whose eyes Jesus Christ hath been evidently set forth, crucified among you?
This only would I learn of you, Received ye the Spirit by the works of the law, or by the hearing of faith? Are ye so foolish? having begun in the Spirit, are ye now made perfect by the flesh? - Galatians 3:1-3
In order to see and enter the Kingdom one must be born again, born from above, born of the Spirit of God and to return to the bondage of the Law is to fall from grace:
Stand fast therefore in the liberty wherewith Christ hath made us free, and be not entangled again with the yoke of bondage. Behold, I Paul say unto you, that if ye be circumcised, Christ shall profit you nothing. For I testify again to every man that is circumcised, that he is a debtor to do the whole law. Christ is become of no effect unto you, whosoever of you are justified by the law; ye are fallen from grace. - Galatians 5:1-4
There are religious edifices constructed by the minds and efforts of men which talk of grace all the while yoking men into bondage to the Law:
Wherefore if ye be dead with Christ from the rudiments of the world, why, as though living in the world, are ye subject to ordinances, (Touch not; taste not; handle not;
Which all are to perish with the using;) after the commandments and doctrines of men?
Which things have indeed a shew of wisdom in will worship, and humility, and neglecting of the body; not in any honour to the satisfying of the flesh. - Colossians 2:20-23
Beware of all who say that something must be added to what Christ has done or who say that some religious activities must be undertaken in order to access or to maintain what Christ has provided. These are patching old garments with new cloth, putting new wine into old bottles; they are vain talkers and deceivers separated from the grace of God.