What are the benefits of purchasing a vehicle over stealing it?

What are the benefits of purchasing a vehicle over stealing it (ie does purchasing a vehicle make it behave like the 3 assigned cars)?

At any point are they essentially the same (ie after saving a stolen car in your garage)?

Some cars are pretty rare or you can't obtain it in other ways than ordering online.

There is no other advantage for buying them in Single player mode.

In single player:

When you buy a car, you own it. You can get insurance on it and successfully store it in your garage.

When you steal a car, you still do not own it and from reports here, you cannot get insurance on it (but I have not tried). When you steal a car, there is a chance the person you steal it from will call the police.

Personal Experience: I have stolen a car from someone while they were in it (pulled them out of the driver seat). While they were running away, I heard them yell, "I am calling the police! Ahh!". Then I proceeded to get a 1 star wanted level.

Note: You can still upgrade the car you have stolen, but if you trash it or lose it, you cannot get it back.

In GTA Online:

Including the items above, a bounty can be placed on you from NPCs. This usually ranges from $1,000 to $9,000... to which other players have to kill you to earn that bounty.