'While away' suggests relaxed, unhurried activities with free time, and it's typically not used for work or stressful contexts. On Sundays, I like to while away the morning with a good book.
Pro Tip 2/3
Time Filler
Use 'while away' when doing something to make the time pass more pleasantly or less noticed, especially during waits. He whiled away the wait for his flight by playing games on his phone.
Pro Tip 3/3
No Significant Outcome
'While away' implies that the activity isn't very productive or doesn't result in anything significant. We whiled away the evening chatting about old times.