If Martians attack, and you have the choice of going with the narrator or his brother, go with the brother.
Neither of them has very much in the way of personality or character – or, heck, name. (Pop Quiz: Name two mental and two physical attributes of the narrator and his brother. It's hard, right? We know some of their hobbies – the narrator likes bicycles and the brother likes boxing – but that's about it.) But at least the narrator's brother has a better track record of getting his companions to safety.
When we compare the narrator and his brother, we find several similarities. For example, they're both in (vaguely) scientific fields. The narrator is science journalist (sort of) and the brother is a medical student. As another example, they both feel some desire to see the Martians killed (1.10.3 and 1.14.4).
At the same time, if we compare the narrator's and the brother's actions towards their companions, the brother clearly comes off as the safer choice for a travel buddy. Whereas the narrator fails to help the artilleryman or the curate, the brother helps Mrs. and Miss Elphinstone reach the coast. And let's not forget that while the narrator (temporarily) loses his wife, the narrator's brother helps save the women (although they also save him from the robbers). In some ways, it seems like the narrator's brother is more heroic.
At least, the brother gets the opportunity to be more heroic. Since the narrator's brother has a bunch of similarities with the narrator, it seems like a lot of what happens to these two characters might just be accidental. For instance, maybe the narrator's brother would also hit the curate if they were stuck in a house together. We'll never know. (But, just to be on the safe side, you should still go with the brother.)
Overall, it seems like the narrator's brother isn't in the book in order to contrast with the narrator. Or, if he is, he's doing a poor job of it.
If the brother isn't here to act as a contrast with the narrator, what's the point of turning the story over to the brother's point of view for three chapters? Well, first, the brother gives a particular, focused view of London and how the big city responds to the invasion. Second, the brother's more heroic and happy story gives us a nice contrast with the narrator's story. We could be wrong about this, but we think the brother gets a few chapters at the end of the first volume just in order to show us another angle of the Martian invasion.