As mentioned previously, application letters and cover letters are not the same. A cover letter identifies an item being sent, the person to whom it is being sent, and the reason for its being sent. A cover letter provides a permanent record of the transmittal for both the writer and the reader.
Remember, you're hoping the person who receives your letter has no memory of your closing at all. The last thing you want is a hiring manager going into an HR meeting with your cover letter in hand, asking the team if they want to meet with "Mr. Kindest Personal Regards."