There's exhaustion, which builds up and is relieved gradually with sleep, and is effectively masked by caffeine. Then there's the feeling of incomplete/interrupted sleep; maybe the quality of sleep was poor somehow, maybe you were woken up by an alarm in the middle of a sleep cycle, but it's painful, cognitively debilitating, and sticks around like a splinter in your brain. Caffeine doesn't help it at all, but it can be completely relieved by a nap in which you succeed in falling completely asleep even just for a few minutes, which caffeine does not prevent from happening.

I don't know how much of that is objectively how it works for everyone, but that's how I understand it.