When you need to update a mod, go to the site and download the new one just like you did originally (only pick the new file). Then go to the TS4 Mods folder, delete the old version, and put the new one in its place. Do all of this before launching TS4.
If you are using TS4 Mod Manager, it's easier to update (imo). You still have to go get the new file, but then when you drag it into the Mod Manager, you'll see two versions. Just click on the old version and then the - (minus) button at the bottom left to delete it.