Other than expenses, they develop much more slowly. As per another commenter a 6 months old pig is practically adult, while it is much longer for most primates.

Primates typically need to be raised in very large colonies to be healthy, and to be able to take from the colony without damaging it, there need to be many many primates (some colonies are in the hundreds).

The Rhesus monkey is a popular one for biomedical testing for a variety of reasons.

Supply? Pigs are both abundant and easily found in the United States. There aren't primate breeding lots. You need to go into a jungle to find them.

You could breed "primates" in all sorts of numbers if you wanted. Especially the smaller species. Some fit in your hand. That's really not the issue.

The issue is that they have very complex social and behavioral demands, and yes, more so than pigs or dogs. It is easy to make a colony of dogs in a laboratory live a very happy life. I've seen it, and I can't see a big difference between a companion dog and a lab dog much of the time. They are just as friendly and outgoing and play all the time. With monkeys that's a lot more difficult...

Which is why research in primates is a lot more restricted and more heavily regulated. You need (more) specialized facilities. You also can't handle them as easily because they will bite, poop or hit (the bigger ones) you.

Down voted because this is absolutely not true. No western research facility uses wild primates this way. They are bred in captivity for a number of reasons, both practical, ethical, and environmental.