Why do source and sink have to be distinct in Ford-Fulkerson?
In a flow network, the source and sink are "special".  The source is defined to have infinite incoming capacity, and the sink is defined to have infinite outgoing capacity -- they are different from all the other networks.  For any node with no incoming edges other than the source, it's true that the flow through that node has to be zero -- but the source is different, because it is special.

Why can't the source and sink be the same node?  Basically, because then you get a different problem with different characteristics.  That other problem is called the circulation problem, and there are algorithms for it.  They are a bit different, though, and your teacher probably doesn't want to confuse you by adding unnecessary complications.