In addition to feeling bitter as a result of Banquo's prophecy—that he would father kings—Macbeth resents Banquo's noble character. He says,
Our fears in Banquo
Stick deep, and in his royalty of nature
Reigns that which would be feared. (3.1.51-53)
Macbeth claims that Banquo is wise and courageous without being reckless. In short, Banquo conducts himself with honor, like Duncan or any good monarch would do. Macbeth fears that Banquo's "royalty of nature" will pose a threat to him, and so it is one reason Macbeth has to order the murder of his former best friend.
Further, Macbeth is bitter as a result of the idea that he will not pass his crown on to his own children (he has none); that he, instead, has saddled his own conscience in order to secure the throne for the descendants of Banquo. He says,
For Banquo’s issue have I filed my mind;
For them the gracious Duncan have I murdered;
Put rancors in the vessel of my peace
Only for them; and mine eternal jewel
Given to the common enemy of man,
To make them kings, the seed of Banquo kings! (3.1.67-74)
Macbeth is angry that he has destroyed his own peace to become king, and that everything he has done will be to make Banquo's children kings. Therefore, he decides to kill Banquo and his only child, Fleance
, so that he can prevent Banquo's line from ever taking the throne. He fails, of course, and James II of England, the king on the throne of England at the time Macbeth was first performed, can actually trace his lineage back to both Duncan and Banquo!