In Julius Caesar, Brutus never gives in to Cassius; he must always have his way. What does this say about Brutus?
In every disagreement between the two, Brutus never gives in to Cassius; he must always have his way.
This is actually a fantastic question, dramatically speaking. It goes to the heart of what makes drama, works meant for performance on the stage, so compelling. Because Julius Caesar is a play, you won't find a definitive answer to this question in the text, you must consult your own opinion (if reading the play) or observe a performance of it and evaluate the actor playing Brutus and his choices of behaviour, to understand why Brutus does what he does.
A play must, by its very design, present all of the characters equally. By this, I don't mean that all of the characters are equally important, but they are all equal in that they are presented to the audience without benefit of, for example, first person narrative of the story, which would clearly give favor to one person's point of view. Even in soliloquy, and a character is not necessarily giving the audience a full picture of what exactly motivates him or her to action. This implies, then, that the reader of the script intended for production upon the stage, is never fully privy to what motivates a character to action. This is part of the work of the actor who will play the role.
And so, with Brutus, I could say that he seems deeply concerned with the fate of Rome, and that it is this concern that motivates him to argue against Cassius, who seems motivated by much more personal, potentially selfish ends. But it is the fact that I can only say that these things "seem" to motivate them, that should give us pause before assigning these or any other reasons for their behaviour in the play. We, as readers, simply can't know for sure.
As a side note, it is interesting to me that your description of Brutus' behaviour suggests a sort of bully -- that he brow-beats Cassius into following his point of view. This is a very interesting take on Brutus, as, simply from reading the play, I perceive Cassius as a real hothead, someone who cannot seem to avoid losing his temper, who really tries to bully others into his point of view, while I see Brutus as very calm and level-headed, not really an "argue-er" at all. To me, it seems that others naturally follow Brutus' lead, not because he has bullied them into it.
So, you can see that, when reading a play, there is much that is left out of how the character is played, making it possible for both you and I to perceive Brutus as quite different. I think both versions are possible within the play, and would depend upon the choices the actor playing the role makes.
For more on Brutus and Cassius and the differing points of view regarding their characters, please follow the links below.