set_crs:
Set the CRS value for the Dataset/DataArray without modifying the dataset/data array.
write_crs:
Write the CRS to the dataset in a CF compliant manner.
The input_crs and inplace arguments are the same; the only difference in signature is that write_crs takes an optional grid_mapping_name.
In short, I'm confused 😉
Maybe consider write_coordinate_system in this issue as well.