dnd 5e – What is the difference between Dragonborn with tails and those with stumps?

Your DM may be indirectly referring to Half-Dragons.

I have not seen any official source for a playable race option, there is a template for a DM to create a half-dragon from other creatures in the 5th Edition SRD.
Half-Dragons DO have a tail, Dragonborn do not.
Dragonborn have dragon lineage, Half-Dragons are the direct offspring of a dragon and another race.

The distinction is pretty vast and is 100% part of the published materials.