Firstly, is there a phrase "get back of"? If there is, it is certainly unusual. The phrase you're thinking of is "get back". "Get back" and "recover" CAN mean the same when get back is a more colloquial expression and recover is more formal or official - difference of register. You would also use recover for different things : abstract things like "recover your composure", and it has, to me, a sense of more action and effort on the part of the "recoverer" than get back.

We often use "recover" when talking about abstract things like health. For example, "I am at the hospital recovering from an injury / I am recovering my sanity," but it can be used for objects too. For example, "I would like to recover my lost wristwatch." Recovery usually happens over a large amount of time. When you say "I want to get back my hair brush from my sister" it is a much shorter amount of time. "Get back" is used often in conversations, but it is not used often in formal writing.

You recover something that you have lost. You an recover lost things. Get backis to start fromwhere youleft last time. One wants to get back to work.Wants get back to normal life etc. Hope this helps.