The usual phrase for the weekdays excluding the weekend is "during the work week" or just "during the week", not "in the week". "In the week" could mean all the days of the week, as for example in,
"There are seven days in the week."
Another way to refer to those days is as "weekday" or "weekdays":
"any day of the week except Sunday, or now usually except Saturday and Sunday"
For "during the week" you can say "on weekdays". That will be understood to exclude the weekend.
There are several idiomatic phrases used with the weekend:
on the weekend
at the weekend
during the weekend
over the weekend
The larger number of phrases for that time may be because the weekend is a special part of the week, and is often the focus of attention.
I think "grammar" refers to patterns of use that are general over a lot of expressions. I don't think these are grammar differences, just customary expressions.