Claudia and Jamie may be on this adventure together, and they're definitely on the same page. But the different ways in which they approach things makes them interesting counterpoints or foils to each other.
Claudia's pretty stubborn. While Jamie can be convinced to go along with things (after all, he accepts the running away business pretty readily), it's usually Claudia's way or the highway. If not, she gets upset (cue the crying, yelling, etc.). Without Jamie there, she probably would have gone berserk, or she would have had a serious meltdown a long time ago.
One prime example is when Claudia states that she won't go home without finding out about Angel. Jamie says: "Well this has been more fun than camp. Even the food's better. There's the difference" (6.77). This whole thing doesn't have as high stakes for Jamie as it does for Claudia. He's just trying to have fun, while she's trying to return as a different person.
On the other hand, Jamie can be reckless. He almost gets caught on multiple occasions, coming out of the bathroom, trying to get his picture taken, and almost joining his class in their museum field trip. Without Claudia to keep him in check, he'd probably have been sent home a long time ago. They're totally different people, but they balance each other out. And that's exactly what they need when they embark on this crazy adventure.