Cathode and Anode Tamer are different versions of the same game, like Pokemon Red and Blue. There are no major differences between the two.
On the New Year's Eve of 1999, a boy named Ryo Akiyama was chatting on a computer in his fathers study with some other people about the Odaiba incident in August. Some people say their parents told them it was terrorist attacks, but one person says that someone named Taichi said it was Digimon. Suddenly, the power goes out, and Agumon appears on the computer screen, asking for Ryo's help. Taichi's Digivice appears on the desk, and Agumon asks him to touch it. He does so, and Ryo is transported to the Digital World.
Agumon tries to explain everything, but Ryo thinks it's all a dream. When a Kuwagamon (Or a Shellmon in Cathode Tamer) attacks Ryo, he realizes that it is real. Agumon defeats Kuwagamon, and takes Ryo to Gennai's house, and Gennai explains everything. After Digimon Adventure, the weakened Mugen Dramon and a wild Chimeramon were near death. The only way they could survive was to jogress. They did, and the result was Millenniumon.
(Although it is never explicitly stated in the games that it was the Dark Master Mugen Dramon, it seems very likely. He seemed to bear a special grudge for Taichi, and it was stated in the games it was a Mugen Dramon Taichi defeated. Millenniumon not only personally saw his imprisonment in this game, in the next, he turned Taichi into stone, while Gennai, who was only two feet away, was only knocked unconsious.)
Millenniumon's power was so great, he was able to take over the Digital World, and cause a time slip, which made the Digimon defeated by the Chosen Children return to life. He then abducted the Chosen Children and their Digimon themselves, and only Agumon escaped to bring help. To defeat Millenniumon, Ryo must chip away at his power by defeating his minions one at a time, and rescue the Chosen Children.
And so Ryo sets off with Agumon, defeating Devimon, Etemon, and Vamdemon, all who make very specific death threats against him. Every time Ryo defeats a villain, a Chosen Child, or sometimes two or three, is saved, and their Digimon are asked to join Ryo. When Ryo arrives on Server, Pico Devimon also taunts him, much as the Adventure one did to the Chosen. (It may even be the same one.)
At last, Ryo arrives as Millenniumon's castle, and fights Etemon and a Chimeramon. But when he defeats the Chimeramon, it only laughs and transforms into it's true form - Millenniumon. But Millenniumon too is defeated, and Taichi is rescued. Now, it's time for Ryo to leave, and all the Chosens' Digimon say farewell - Patamon likes him second to only to Takeru, and so on. Agumon and Ryo hug, and Ryo is transported back to the Real World, and as he passes in the sky, every Digimon on File Island says goodbye.
Ryo finds himself back inside his house, with the power still out. He and his parents talk, and they note how much he's changed in what they see as the short time they were apart. However, that's not the last of Millenniumon...