The major differences between the two is that Half-Dragons have a rechargeable breath weapon in 5ed, where dragonborn have to wait for a rest in order to get theirs back, Also Half-Dragons have better vision in the dark. So it it pretty much the minor details that make the two different.
Dragonborn was created in 3.5 not 4 edition, in 3.5 in order to become a dragonborn, you had to share the same ideals as Bahamut but didn't have to be Lawful Good, 100 GP spent to make a shell made out of good dragon scales. After the ritual any race would get a +2 Con and -2 Dex on top of their...