How different are the following two sentences:
I am not allowed to sleep during the day. I am not allowed to sleep in the day.
The sentence with during is the correct one; the other one is incorrect.
The preposition during generally refers to a time interval in its entirety: in your example, the entire day (meaning "daytime") is a time interval in which sleep is forbidden.
The preposition in is sometimes useable for referring to a small portion of a time interval. It doesn't work in your example, and I apologize but I can't explain why. It may be that it is grammatically correct, but "in the day" just isn't generally used, and doesn't sound natural.
Some examples using in that are correct and natural:
I will bring you to the doctor's office in the morning.
The fireflies will come out in the summertime.
We'll be on vacation in November.
This last example is a good one for the in vs. during contrast. If you say:
We'll be on vacation during November.
then you mean you'll be on vacation for the entire month, not just for a few days.