How is the apparition in Luke chapter one different from other apparitions, such as in Lourdes?

This is one that has an actual biblical backing to it. Remember 2 Timothy 3:16 declares “All Scripture is inspired by God and is useful to teach us what is true and to make us realize what is wrong in our lives. It corrects us when we are wrong and teaches us to do what is right.”

God inspired Luke to write about Mary being visited by Gabriel and than a few months later Joseph. These we know to be true for the Word of God says they happened. However, we don’t have that kind of backing for other people claiming to see visions, and hear from God. Instead they must be proved, or be ignored. Colossians 2:18-19 mentions people having “visions” which is of their own imagination, and he condemned them quite strongly in said verses “Don’t let anyone condemn you by insisting on pious self-denial or the worship of angels, saying they have had visions about these things. Their sinful minds have made them proud, and they are not connected to Christ, the head of the body. For he holds the whole body together with its joints and ligaments, and it grows as God nourishes it.”

They are too many people these days who are seeking a spirituality that goes further than the Bible, and will willingly open themselves up to deception, such as Bethel Redding in California, who are bringing occultism into their beliefs, because they are not satisfied with God and His word, and instead seek after experiences instead of Biblical Accuracy.

Edit: To the guy who mentioned “Why even bring up Bethel” I did because Bethel is a perfect example of those who are seeking emotional experience, who are seeking a mystical approach to things, who are consumed in their lust for symbols and signs, to the point that they create their own. AKA the so-called glory cloud of Bethel. We should be very careful about what we trust, a vision of Mary or of an Angel has no standing over the word of God. Paul in Galatians 1:8-9 writes “But even if we or an angel from heaven should preach to you a gospel contrary to the one we preached to you, let him be accursed. As we have said before, so now I say again: If anyone is preaching to you a gospel contrary to the one you received, let him be accursed.”

Paul also says this in 1 Corinthians 11:13-15 “For such men are false apostles, deceitful workmen, disguising themselves as apostles of Christ. And no wonder, for even Satan disguises himself as an angel of light. So it is no surprise if his servants, also, disguise themselves as servants of righteousness. Their end will correspond to their deeds.”

There is great danger in putting theological significance in places or people outside of Scripture.

How is the apparition in Luke chapter one different from other apparitions, such as Lourdes?

Well, first of all, with the death of the last Apostle also ended what the Church calls Public Revelation. That means at the death of the Apostle St. John ended the time that any biblical texts could be added to our canon.

Any apparitions after this are considered to be considered as Private Revelation and as such the Church does not oblige her faithful to believe in.

Public and Private Revelation

Public revelation is what we have in Scripture and Tradition. It was completed, finished, when the last Apostle died and the New Testament was finished. So there is no more until Christ returns at the end. In this area the Church has His promise of providential protection in teaching.

Even though there is no new public revelation, the Church can progress in deepened understanding of the original deposit of faith--thus the Immaculate Conception, for example, was not mentioned in the first centuries, was even denied by many in Middle Ages, but could be defined in 1854. This progress is the result of the growing light of the Holy Spirit. At the Last Supper Jesus promised Him to lead the Church into all truth.

Private Revelation is all else. The word private is poor, but usual. Even Fatima, addressed to the world, is private. But there is a great difference. The Church does not have the providential protection in matters of private revelation. Ordinarily the decision of the local Bishop is final on authenticity of a revelation. Yet we would not have to believe any decision on private revelation--though we must obey a command, if a Bishop gives such, not to go to the place of a an alleged revelation. In obeying, we do not lose any graces. Christ saved the world by obedience--cf. Rom. 5:19. St. Margaret Mary says He told her: "Not only do I desire that you should do what your Superior commands, but also that you should do nothing of all that I order without their consent. I love obedience, and without it no one can please me."

The most the Church can do on a private revelation is: 1) say it does not clash with public revelation. If it did, that part of it would be out. 2) Say it seems to deserve human acceptance--that is in contrast to something accepted on the divine virtue of faith, which comes into play only in the area of public revelation.

The Church does not have to investigate the apparition of St. Gabriel to Mary because it is clearly part of Scripture.

All another apparitions are investigated to make such there are no errs concerning faith or morals.

You can read the Vatican Norms Regarding the Manner of Proceeding in the Discernment of Presumed Apparitions or Revelations at your leisure.