Use 'worn out' to describe extreme physical or mental tiredness, not just general sleepiness. After the marathon, every runner looked completely worn out.
Pro Tip 2/3
Emotional Emphasis
'Worn out' can emphasize feelings of being overwhelmed or fed up with a situation. I'm so worn out by these constant arguments.
Pro Tip 3/3
Colloquial Tone
Mostly used in informal conversations; may be less appropriate for formal writing. I can't go out tonight, I'm just worn out.