This is a difficult question, because it concerns internal motivations. It is even difficult to judge our own motivations. However, we can surmise. Let me focus on Brutus. When it comes to Brutus, we must look at past Roman history. When we do this, we read of another Brutus that acted on behalf of the Roman people. When Tarquinius Superbus was the king of Rome, some Roman believed that they must get rid of the corrupt king. What they wanted was a government that was more equal and protected the people. This was the beginning of the Republic.
When we come to Caesar, we read of another Brutus. This Brutus was probably motivated by duty, a love for the Republic, a sense of honor that stemmed from the other Brutus. When he realized that Caesar stepped the boundaries of the Republic, he wanted to reestablish the Republic, like his noble ancestor.
For Cassius, the only real principle that governs him is loyalty. During Rome's struggle for leadership between Caesar and Pompey, Cassius sided with Pompey and hates Caesar's victory and ensuing power grabbing. Other than a sense of loyalty to Pompey, Cassius wants revenge for his loss and is jealous over Caesar's popularity with the people and many members of the Senate.
Brutus, on the other hand, is motivated by honor and duty. As a senator and as the member of a family that fought to establish the Roman Republic, Brutus feels obligated to hinder any one person from becoming too powerful and restoring the monarchy/dictatorship that once governed Rome. His speech at Caesar's funeral represents this motivation when he says, "Not that I loved Caesar less, but that I loved Rome more." At the play's end, Antony makes a distinction between Cassius's and Brutus's motives by singling out Brutus and praising him as the most noble Roman of all (posthumously).