What's the difference between that and who?
The standard answer says we use who when we refer to a person; we use that when we refer to things. But again, it's a rule broken as often as it's followed.
Many people include animals or any living creature in the who category, which traditionalists dislike. They insist on using who only for animals with names. That seems arbitrary to me.
It's helpful to remember the rule (especially if you have too many that expressions in your prose.) My assumption is that most readers don't know or care about the difference.