It seems that I slept during the holiday can be translated into two ways:
Mi dormis en la festotago.
Mi dormis dum la festotago.
Do they both mean the same?
Your two sentences mean different things.
Dum implies duration. En suggests that something happened within the period.
At some point you slept.
You spent your festival day sleeping.
PMEG has a paragraph about this. http://bertilow.com/pmeg/gramatiko/rolmontriloj/rolvortetoj/lokaj_rolvortetoj/en.html#i-sb8
Dum shows a duration (over which the action took place). En can show one of two things
You can also have temporal tra and por.
Mi dormis tra la festotago - almost the same as with dum, but with emphasis on sleeping through the whole period.
Mi ripozis por unu tago - I took a break which lasted one day.