A handle is generally used for cleaning tools like broom or mops to help the cleaner have a stronger grip and apply more force while they are cleaning. The handle can also allow the user to have a greater reach by being extendable.
A brace is commonly used to help with a broom or mops stability. This generally happens when you have a longer handle and you start to notice that the handle is beginning to slip out. The brace will make sure that the handle and cleaning head stay attached and stable during its usage.