"He is tired"
-me- indicates an action or change of state occuring in the past that has present relevance. It's a bit like saying "He has become tired." (With has become, just like -me-, the implication is that he is still tired.)
-li- has no such indication of present relevance. If you say alichoka, you're just saying he became/got tired in the past without saying anything about his present state (eg. maybe it was years ago and he has obviously slept many times since then).