Relative Clauses

Do you know the difference between these two sentences?

My brother, who lives in Nottingham, is a doctor.
My brother who lives in Nottingham is a doctor.

Here's a clue! How many brothers do I have?
​Think about your answer then click "read more" to check.

In the first sentence, I have one brother and he is a doctor. (By the way, he lives in Nottingham - extra information which is not important.) The commas are like brackets.

In the second sentence, I have more than one brother, and the you probably know this. "Which brother is a doctor?" you ask! The one who lives in Nottingham. I have another brother who lives somewhere else. My brother who lives in Nottingham is a doctor but my brother who lives in London is a lawyer. The Nottingham brother is a doctor and the the London brother is a lawyer.

One more thing - is it possible to replace "who" with "that"? In the first sentence, I can't but in the second sentence I can. 
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My brother that lives in Nottingham is a doctor.

By the way, I have do have one brother but the rest isn't true! My brother, who lives in Southwell, is a computer programmer.