What was Montanism, and why was it rejected?

Montanus (as Epiphanius reports) fl. c. 170 ADOracle, quoted in Epiphanius, Panarion 48.11.1

"I am the Lord God almighty, dwelling in a man."

Epiphanius cites this as proof that Montanus confused himself with God. Montanists took it as God speaking through the prophet.

Maximilla (Montanist prophetess, quoted by Eusebius) d. c. 179 ADOracle, quoted in Eusebius, Church History 5.16.17

"I am driven away like a wolf from the sheep. I am not a wolf: I am word and spirit and power."

The anonymous writer demands that the spirit in Maximilla prove its power before the bishops who tried to examine it.

Anonymous anti-Montanist (quoted by Eusebius)Treatise, quoted in Eusebius, Church History 5.16.7

There is said to be a village in Mysia in Phrygia called Ardabau. There, they say, one of the new believers, named Montanus, when Gratus was proconsul of Asia, in his boundless desire for first place gave the adversary a way into himself. He was carried away in spirit, suddenly fell into a kind of possession and ecstasy, raved, and began to babble and speak strange things, prophesying contrary to the custom of the Church handed down by tradition and succession from the beginning.

The charge was ecstatic, uncontrolled prophecy against the Church's tradition, not prophecy as such. Synods in Asia examined and condemned it.

Apollonius (quoted by Eusebius)Against the Montanists, quoted in Eusebius, Church History 5.18

For we shall show that the men called prophets and martyrs among them collect small change not only from the rich, but also from the poor, and orphans, and widows.

Apollonius turns to the fruits test. In the same passage he says Montanus appointed money collectors and paid salaries to the men who preached his word, and that a prophetess had accepted gold, silver and costly clothing. A prophet who takes money from widows and orphans, he argues, fails the test the Lord himself gave.

Tertullian c. 155–c. 220 ADOn Fasting, against the Psychics 1.1–3

I would be surprised at these psychics if they were held only by lust, which makes them marry more than once, and not also torn by gluttony, which makes them hate fasts. ... I recognize, then, an animal faith devoted to the flesh ... so that it rightly accuses the spiritual discipline, its rival, in this matter of continence too.

Tertullian, now a Montanist, calls Catholics "psychics" (unspiritual) for rejecting the New Prophecy's stricter fasts and ban on second marriage.

Epiphanius of Salamis c. 310/320–403 ADPanarion 48

They boast that they have a certain Montanus as prophet, and Priscilla and Maximilla as prophetesses; paying attention to them, they went astray.

Epiphanius opens his account of the Phrygians by naming the three figures at the head of the movement. He writes two centuries later, as a hostile heresy hunter, and counts their attention to these prophets as the error itself.

Epiphanius of Salamis c. 310/320–403 ADPanarion 49.1

In the form of a woman, she says, clothed in a bright robe, Christ came to me and put wisdom in me and revealed to me that this place is holy and that here Jerusalem comes down from heaven.

Epiphanius writes two centuries after the events, as a hostile heresy hunter, and the vision of Priscilla (or Quintilla) at Pepuza is reported only by opponents of the movement. He tells it against the Quintillians, who treated Pepuza as the holy place where the heavenly Jerusalem would come down.