I would have thought that both were different, but grammatically correct, with almost identical meaning.
"A properly written sentence" is a sentence that is properly written, and "a proper written sentence" is a written sentence that is proper.
But now I have achieved semantic saturation, and am not sure anymore.
[1] https://owl.purdue.edu/owl/general_writing/grammar/how_to_us...
The way I read it, the choice of "proper written sentence" gives it the flavour of "Proper. Written. Sentence" - as in, with an annoyed emphasis.
As always, you can apply rules to English if you like, but sometimes they are optional.