A letter of application for a job is the same as a cover letter. Often recruiters and hiring managers will use the terms interchangeably to refer to a letter sent to give a hiring manager more information about your qualifications.
But application letters (or letters of application) for academic programs and scholarships differ from cover letters. Compared to a cover letter, an application letter for academic purposes is:
tailored to the academic program or scholarship’s goals, rather than a specific job
often longer (acceptable to be 1–2 pages)
accompanied by other application materials (e.g., your transcript, resume for college, personal statement)