AWOL vs. desertion
From that Courts Martial manual--basically the distinction is intent. Desertion is absence "with intent to remain away therefrom permanently" or "with intent to avoid hazardous duty or to shirk important service." Generally, "the accused’s absence was terminated by apprehension." He didn't come back voluntarily.
>Desertion with intent to remain away permanently is complete when the person absents himself or herself without authority from his or her unit, organization, or place of duty, with the intent to remain away thereform permanently. A prompt repentance and return, while material in extenuation, is no defense. It is not necessary that the person be absent entirely from military jurisdiction and control.
AWOL basically "the guy didn't show up when he was supposed to" but without the intent that it be permanent. "any member of the armed forces is through the member’s own fault not at the place where the member is required to be at a prescribed time...Specific intent is not an element of unauthorized absence." There are some step ups in punishments after 3 and 30 days of absence.
You can read the whole 10 MB thing if you need an insomnia cure. MANUAL FOR COURTS-MARTIAL UNITED STATES (2005 EDITION)
