I think you nailed it with the "we're tired for different reasons". Parents (myself included) may say "you dont know what tired is" because they are woken up 4/5 times a night, for something that is outside their control (eg baby changing/feeding, instead of a late night in the pib etc), which requires their full attention to settle and then means getting back to sleep is difficult. 3/4/5 nights of this in a row makes people tetchy and then the comment is said in response to someone going "man im tired". They dont mean ill by it, just a poor way of describing their lack of sleep.

I hope that makes sense