No, the company you paid it to will not directly send you a check back. But they do reduce the "payoff amount" by the amount of your payment.
Check your paperwork with the dealer you bought your new car from. It will have a "payoff amount," by the dealer likely won't cut a check to pay off the loan until several days later.
If you determine that the payoff amount is now lower than what your paperwork says, go back to the dealer and show that to them. You have a good basis for asking that the loan amount be reduced by the difference. Demand it of them.
If they have already made the payoff, call the lender on your old car and determine how much was paid off. If you can demonstrate to the dealer that they paid off less than the amount agreed, they owe it to you you make up the difference. But they likely won't unless you ask them.