Both Macbeth and Banquo, military leaders in Scotland, have thus far, by all indications, lived their lives in honor and service to their king. This service was the focus of their ambition until the encounter with the Three Weird Sisters.
In the prophecy, Macbeth is predicted to become king, while Banquo is told that it is his children who will ascend the throne. The nature of these prophecies perhaps regulate the reactions of the two different men.
Banquo is not told anything about himself, but only his children. Therefore, he does not have an immediate vested interested in taking action to fulfill the prophecy. Macbeth, however, does have an interest and takes upon himself the responsibility to bring it to fulfillment, rather than let Fate take its course.
Banquo, in discussing the prophecy, expresses doubts as to the validity of such a thing, coming as it does from witches. While he admits that it might contain some element of truth, he is wary that its way is to lead them to evil, rather than success. In this, Banquo shows a level of wisdom and insight that Macbeth does not possess.
Banquo's refusal to take action in fulfilling the prophecy conflicts with Macbeth's choice to do so, aided by his wife. This puts Banquo in a dangerous position, especially since, according to the prophecy, it will be Banquo's children, not the children of Macbeth, who will succeed Macbeth. Unsure of this aspect, Macbeth is taking no chances and murders Banquo as well as Duncan.