"The rhesus macaque is pretty much the closest thing we have to humans" -- why?
Short answer:

Chimpanzees are indeed closer, but perhaps...too close. Almost all invasive research on non-human great apes has halted for some time over ethical concerns.

Rhesus macaques are the main non-human primate still used in research (some others are also used, like marmosets). "Pretty much the closest" is in reference to which species are available to scientists as a model organism; I assume the researcher quoted included "pretty much" as a bit of a hedge, acknowledging that chimpanzees are closest while not being available for vaccine research. Even then, many countries have limited research using all primates and do not use macaques anymore, either.



Longer answer:

The primates include great apes like gorillas, orangutans, and our closest non-human relatives: chimpanzees. They also include more distant relatives, like Old-world and New-world monkeys, and even more distant lemurs.

Many humans (including influential organizations) have decided that chimpanzees and other apes are too closely related to humans to be eligible for certain types of research for ethical reasons. It has been deemed more ethical to use human subjects, who can consent to participation in studies, or to use other model animals. Apes are also difficult as model organisms: they develop slowly, grow large, and have long life spans with few offspring; all these factors make them expensive research subjects. There is no one biological fact for this distinction between apes and other monkeys, besides their closer relationship to us and the perception that they have more advanced intellectual faculties. Further discussion is outside the scope of this answer.

Old-world monkeys, including rhesus macaques, are the next-closest grouping of primates to the human lineage. Macaque monkeys live and breed fairly well in captivity, and are used in research when other model organisms like mice and rats are unsuitable (like when an aspect of their biology is not sufficiently similar to humans).

Rodents can also be models for vaccine research, but their distance from humans and differing response to certain pathogens may make them less useful for measuring efficacy. Therefore, close relatives like rhesus macaques have been used in past vaccine research.

There is no perfect model organism; even humans aren't always great models for other humans, when they vary by age, sex, pre-existing conditions, etc. There is a balance in risk and ethics in choosing model organisms for research: simpler animals like mice and rats are typically used at earlier stages of research. Primates are used when rodent models don't correlate well enough with humans or when there is a large risk to human trials.



Some references:

https://en.wikipedia.org/wiki/Animal_testing_on_non-human_primates

https://www.nih.gov/about-nih/who-we-are/nih-director/statements/nih-will-no-longer-support-biomedical-research-chimpanzees

https://www.nature.com/subjects/rhesus-macaque

https://www.sciencemag.org/news/2018/11/record-number-monkeys-being-used-us-research

Nath, B. M., Schumann, K. E., &amp; Boyer, J. D. (2000). The chimpanzee and other non-human-primate models in HIV-1 vaccine research. Trends in microbiology, 8(9), 426-431.