Either are grammatically correct, they just mean different things. Preference doesn't really factor into it - either work, it would depend on what meaning you wanted to convey.
To "get" something means to obtain it, bring something into your possession. Idiomatically we do use it to describe the action of collecting something. However, it can also be used when receiving something. For example:
I got my permit today.
This could mean that you went and collected it from somewhere or equally that it arrived in the mail.
"Collected" in this context specifically describes the action of you calling at a place to obtain something and then taking it away.
To sum up then - both could idiomatically mean the same thing, but "get" allows for more meanings while "collect" is very specific.
Other expressions that could be used to show that you "collected" something include:
I picked up my permit today.
I went and got my permit today.