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.
Formal communication is on the wane in modern life, but there are still times when it's the only correct way to reinforce a connection or convey information. If you're applying for a job, looking for a recommendation, or expanding your network, err on the side of formality. Don't let the slightly archaic feel of a formal business letter tempt you into using flowery, outdated language.