Difference between the sentences

If I had only one brother named James I would write it with the two commas as in your first example, and if I had more than one brother I would write it without commas as in your second sentence. But I wouldn't be surprised to see people using either form to suit either case.

Either sentence is correct.

I personally do not think the first implies just one brother. I could for example write, "My brother, James, is clever, but my brother, George, is a clod."