They are basically the same "free of charge" is written more fully.

I’d say the second one is more formal and it’s only limited to payments. “Free“ of course has other meanings.

They are really very similar, and you can usually use them interchangeably. However, in my mind, "for free" is a little bit more informal/casual. For example, "She is teaching English for free." & "She is teaching English free of charge." Both mean the same thing ("She is not being paid to teach English") but 'free of charge' sounds more professional. Hope that helps! They really are very similar, so you shouldn't worry about when to use them.