My sister, Jane, and I went to the museum.
It is just the two of them, no other sisters. While it is great that we now know the sister’s name, the meaning of the sentence does not change.
My sister Jane and I went to the museum.
Oh my, hopefully the other sister(s) were not jealous. By leaving out the commas, Jane is promoted to essential information and indicates that there are additional sisters.
My wife, Mrs. Jones, is on her way.
Great, hopefully she’ll get here soon!
My wife Mrs. Jones is on her way.
Really now? Should this be a case for the law or the comma police?