The movie La jetée has already the story of 12 Monkeys, including also key-settings like the airport and the mall. But there is one big difference in 12 Monkeys to the older movie: Terry Gilliams included Jeffrey Goines, who plays an important role in his movie. Why did he include Jeffrey?
Interestingly Gilliam did not write the screenplay for 12 Monkeys, it is a rare example of a Gilliam movie that he did not start but was brought on-board after the rights had been picked up by the studio and a screenplay had been written by David and Janet Peoples.
That said, La jetée is 28 minutes long. I imagine that the writers could not sustain the story for 90+ minutes with just the main characters of the Narrator and The Woman, so introduced the both the disease as the cause of the 'end of the world' rather than a nuclear war, and Jeffrey as a character for Cole to interact with and start to obsess about stopping him.
I also imagine that when they came to rewrite the story, that a plot of a disease being deliberately released by eco-terrorists plays into the fears of people in the 90's more than people in the 60's when the original was made with a nuclear war story. This alternative back-story also makes it plausible for Cole to attempt to stop the apocalypse from happening as a driver for a lot of the action.
Really 12 Monkeys is not a remake of La jetée, as it says in the credits, it was inspired by the original. In La jetée the narrator is also sent into the future where he brings back technology to help his broken society and is not killed by present-day cops at the Airport.