In the first, who lives in Paris is an integrated relative clause. In the second, it’s a supplementary relative clause. Those, at least, are the terms used in ‘The Cambridge Grammar of the English Language’. Such clauses are also known as ‘defining / non-defining’ and ‘restrictive / non-restrictive’.
The first sentence suggests that the speaker has more than one friend. It allows us to believe that the speaker may have friends who live in New York, London and Berlin, as well as the one who lives in Paris. From the second, we understand that the speaker has only one friend and that that friend lives in Paris. It is supplementary, because we can remove the clause and still be left with a viable sentence. A supplementary relative clause merely provides additional information.
In writing, it is the convention to place commas around a supplementary relative clause, as shown. It is possible to indicate more clearly when a relative clause is integrated, rather than supplementary, by substituting that for who (or which). A supplementary relative clause is not normally introduced by that. In speech, a supplementary relative clause is indicated by intonation and by a slight pause at the start and end of the relative clause.
OK, OK. Sentence 2 envisages only one friend in the context of the sentence. Here’s how the clauses might occur for real.
I know people in several European cities, and I suppose they’re all my
friends, really. The one in Berlin, I don’t know her quite so well as
the others, but my friend who lives in Paris, now she’s really great
So, this friend of mine told me what happened to her last year. I was
really amazed, but tell me what you think. See, my friend, who lives in Paris,
was on her way to work one morning when she saw what she thought was
an accident in the Rue de Rivoli . . .