There are roughly two major schools of thought on the subject of time travel in (science) fiction. One is that a timeline is "flexible" and can be changed. When a certain outcome is undesirable, one needs to travel back to a point in the past, and change whatever caused the undesirable outcome to happen. For instance, in order to save your mother from being run over by a car, you travel back in time so that you can push her out of the car's way and save her life. Most popular movies about time-travel adhere to this principle (e.g. Back to the Future (1985), Donnie Darko (2001) and The Butterfly Effect) (2004).
Another theory is that the timeline is "fixed" and can't be changed. The present is already the result of the past plus all attempts to try and manipulate the present. This implies that whatever you try to prevent by traveling back in time will be unsuccessful, or, worse, instrumental in causing whatever you tried to prevent. In case of the mother example, according to this theory, you can travel back in time to a moment before she was hit, but any action you might take will not be successful in saving her. Possible scenarios are that you don't make it in time for some reason: perhaps in your hurry, you are hit by a car yourself, or you could shout at her to watch out for the car, thereby distracting her from the traffic, and thus causing her death. Movies following this principle are Sphere (1998), Harry Potter and the Prisoner of Azkaban (2004), The Time Machine (1960), and Twelve Monkeys.
Cole says that the future can't be changed, which proves to be true. Throughout the movie we see and hear about things that are said to happen in the future, and they all inevitably happen. Of course, the main event is that the global virus infection occurs. Cole's mission is not to prevent it, because the scientists know he can't; the very fact that the infection happened is evidence that it could not be prevented anyway. Furthermore, by traveling back into the past, Cole sets in motion a chain of events that lead to the very future he comes from, e.g. the mysterious recording by Kathryn to the future, in which she identifies the Army of the Twelve Monkeys as the source of the virus. Cole and the scientists hear the recording and Cole is sent back in time to investigate; however, because of Cole's interactions with Kathryn Railly, he causes Kathryn to make this recording. In essence, the events that trigger him to travel through time actually cause them, which is known as a time paradox.
The fact that the future is set does not prevent Cole from trying to change it anyway. It is in his nature to try, simply because he does not know the exact details of what caused that particular future to happen and what his failure in trying was. For instance, it is known in the future that there was an outbreak of a virus, but not who or what started it. After the Army of the Twelve Monkeys appeared to be a dead end, Cole and Kathryn identify the real perpetrator at the airport, and try to stop him. This ends in Cole's death, which is witnessed by the young Cole, explaining the strange dreams he always had. It had never occurred to the young Cole that he actually saw himself dying, so the adult Cole never knew what fate would await him if he would try to prevent the virus outbreak.