A datetime fields saves a date time. How you see that date time is a result of the tool you're using to inspect the data, whether it is Management Studio, or your own software that's printing something from the database.
I strongly recommend keeping it as a datetime field. This will allow you to do date-related operations, such as subtractions and comparisons. If you want to change how your users see the date, then format your date at the presentation layer.
What's happening in the code you've posted is that you're setting the value of A
to the same date that it already is. The fact that you're setting that value by means of a string in another format has no relation, SQL server will always have to parse your string input into a date that it can understand. This is why you're not getting an error message. The operation is working, only it's not changing anything.