Well, first of all, no one's ever been offended by nor has a fantastic movie ever spoofed the line "What do you mean 'You persons'?!?".

People = a collective or indeterminate number - "the French people"

Persons = a group of individuals with a known quantity - "four persons from France"

Being said, persons is generally accepted as archaic outside of legal and some more formal print media. I think the AP stylebook says that people is preferred in all situations now.

People is used when there is one group you are referencing as a whole and you don't need or want to distinguish or highlight the distinctions between the individuals in that group.

Persons is used when you have been talking about and making distinctions between individuals within a group but want to reference the group as a whole.

For this reason persons is used most frequently in legalese.

Persons of interest identifies that there are multiple people the cops are looking into. However they may not be accomplices. It could be the wife OR the gardener that killed the guy. However the cops don't think they were in on it together.

Hence they are PERSONs of interest not PEOPLE of interest.

If there are 4 interested parties you are looking at selling your car to it is more grammatically correct to say

There are four persons making an offer on my car.

The statement:

There are four people interested in buying my car

implies that all four people want to buy it together in one transaction and then share your car as coowners

However nowadays outside of legal circles this distinction is rarely made.

Dear God I am Cliff Claven

Yeah but you do have excellent person skills