1. The author never claims to be an apostle
One response would be that the churches to which he writes would know which John was writing to them, so he did not need to specify which John he was, nor did he need to emphasize his apostolicity.
2. The style and vocabulary of Revelation is markedly different from the Gospel of John and the letters of John.
This is indeed a strong argument. But one response would be that the vastly different content explains the change in style and vocabulary.
3. The quality of Greek grammar is far inferior to that of the Gospel and letters of John
This is perhaps the strongest argument. But some respond by saying that John is writing in his second language from a prison island with no amanuensis (secretary) to clean up his Greek. Others believe that the author writes deliberately in slightly choppy Greek in order to slow down his readers and force them to notice the details in his work.
4. The theology of Revelation can be said to differ from the theology of the Gospel and letters of John
Some say that John’s God is love, but Revelation’s God is judgment. This is a naïve idea, as in the entire Bible we see both the love and the justice of God.
Some say Christ is portrayed as revealer and redeemer in John, but as warrior and ruler in Revelation. Again, there is no necessary tension between these various roles of Jesus.
Some say that John’s Gospel focuses on ‘realized eschatology’ (the changes the gospel makes in people here and now) whereas Revelation focuses on future eschatology. But this contrast is unfair. John’s Gospel is filled with references to future eschatology.
5. There is a tradition that claims that John the apostle died early as a martyr. He would not have lived long enough to write Revelation.
But many from the early church believed that John the apostle wrote, and that he wrote in the time of the emperor Domitian, toward the end of the first century. So the tradition was far from universally accepted.
6. Revelation names its author, whereas the Gospel of John is anonymous, written by “the disciple whom Jesus loved”.
This is indeed a surprising difference, but does not rule out the possibility that the author wanted, out of modesty, to exclude his name from the Gospel but felt it important to include his name in Revelation, to substantiate the prophecy he received from Jesus.
7. Nothing in Revelation indicates that the author knew Jesus personally. This is surprising if the book was written by one of the twelve apostles.
The author is reporting a vision, not writing a treatise. He would have no reason to mention whether or not he had met Jesus during the latter’s earthly lifetime. And anyway, why must we assume the apostles would feel the need to mention their having met Jesus in every single work that they wrote?
8. Some people in the first centuries of the Christian church (Gaius, Dionysius, Marcion) did not believe John was the author.
Gaius and Dionysius, both from the third century, had a theological motive for not wanting John the apostle to be the author of Revelation: they wanted to stop the millenarian movements that used Revelation 20 for support. Marcion was a heretic that rejected many portions of the New Testament that didn’t support his unorthodox beliefs.
Sources: David Aune, Word Biblical Commentary: Revelation 1-5. (Waco, TX: Word, 1997); Carson, Moo and Morris, An Introduction to the New Testament. (Grand Rapids: Zondervan, 1992)