One of the main areas that remains the same is the profile folder it's self. If Mary Roe married John Doe and wanted to change her name, one way to do this is:
First Create a new profile Mary Doe
Log into that profile at least once.
Reboot the computer and log in as an administrator that is neither Mary Roe or Mary Doe. While logged in as administrator you need to make sure you can see all hidden files. Then click on the the Folder c:\Users\Mary Roe\ and select all the contents. You can do this by clicking on a single file, then do a Ctrl-A to select all files, then do a Ctrl-C to put the files in the copy mode. Go to the new folder c:\users\Mary Doe\ and right click in that folder and paste. You will now get a confirmation to overwrite files and/folders. put a check mark in the "all" section. You will get that normally 3 times. After the copy is complete. Mark sure the new profile is in the Administrators group and the remote desktop group. Reboot and log back in as Mary Doe. The profile should look the same as Mary Roe used to look when you log in. RDP to this account should work also. I would wait for a short while before deleting the Mary Roe profile in case you made a mistake. When you are ready to remove Mary Roe, don't just delete the folder, but instead go to system properties\ advanced tab \ profiles button. Select the profile there and remove. Once removed it can not be restored, so use caution.
I have seen techs take a quick shortcut and just rename the folder. But that does not change registry keys and while it is very quick, I have also seen it fail.
There are areas of caution, one that I have seen is that Outlook records the whole path, so if you have a Outlook PST file in Mary Roe, after copying the profile to Mary Doe, Outlook may still point to Mary Roe. If you are worried about this you can create a backup of the folder on external drive then remove the Mary Roe folder from the PC and see if anything points to it.
Over the years I have seen several ways to do the above, but the one I explained worked for me every time. And I have done it hundreds, if not thousands of times. Good Luck!