Why is the South American squirrel monkey a invasive species?

We're not 100% sure that they are. A non-native is not necessarily an invasive. Invasives are those species that, once introduced to a new range, outcompete their native counterparts and other species by modifying habitat or over-utilizing food resources. We see it with snakeheads, pythons, etc in the U.S. Mexican gray squirrels are known to occur in Florida but haven't expanded their range since introduction. There may be unknown pockets of them around, but if they were truly invasive they would likely be more common then they currently are and would no doubt drum up some concern in the scientific community.