To add on what Noodles said, I find it easier to think of the two as such:
persons is a group of individuals, while people is a faceless group where the emphasis is on the group, not the people in the group.
And yes, most often used in a conditional statement. But (as Noodles said, and as illustrated in the last part of my previous sentence) these days, persons is now interchangeable with people, and very rarely used indeed.