The species we use for research are called "model species". We use these species for a combination of reasons. Part of it is because they have some relevant connection to human biology (or other kinds of biology, depending on what we are studying). But just as important, if not more so, are other factors. Things like "how difficult/expensive is this animal to keep in captivity", "how much do we know about this animal in general", "what are the ethics of working with this animal", "how easy is this animal to work with in a lab setting", "how many other scientists work with this animal", and "how easy is it to get these animals".

It's not that rhesus monkeys and baboons are particularly more human-like than other old-world monkeys. But they are widely available, reasonably easy to keep (for a primate), and have had plenty of existing research done on them, which means their care and biology is understood already. All this makes them easier to use than some other species.

You can see similar patterns with lab rats and mice, zebrafish, fruit flies, C. elegans, and other model organisms. It's not that they are especially different from their relatives, but they are widely used, cheap, easy and fast to raise.

Contrasting, say, rhesus monkeys with chimpanzees....chimps are bigger, more intelligent and harder to keep suitably in captivity (so more ethical issues), they reproduce more slowly, and they are endangered. Even though they are more humanlike in their biology, working with them is kind of a nightmare. It's more expensive, you need bigger facilities, you aren't going to get ethics board sign offs as easily, and it's not easy to source chimps. And especially if you are studying the immune system, other animals are pretty close. The marginal benefit of working with chimps just doesn't outweigh the extra difficulty. Really the only time you see research done on chimps is when there's some particular reason you can't use other primates (or non-primates).