With the news breaking today regarding the Plantagenet DNA, it seems that people are already confusing paternity with seniority. While both of these concepts come into play regarding inheritance (in this case, the throne of England), they are not the same thing.
One of the things that researchers had hoped to discover was the elusive Plantagenet Y-Chromosome. This chromosome can only be passed down from father to son. (Conversely, the mtDNA is passed from a mother to all of her children, but only her daughters will pass it along to their children. See http://sunnesandroses.blogspot.com/2014/03/joan-beaufort-her-many-descendants.html )
In order to determine conclusively what the Y-chromosome for the Plantagenets was, there needed to be a genetic match between the DNA of Richard III & that of the male descendants of John Beaufort, the 1st Earl of Somerset, the son of John of Gaunt. Unfortunately, there was no match. When & where the break (or breaks) in the line occurred are a mystery & I'm sure they will be the source of endless speculation. I'm already doing some preliminary research in this area.
While Richard III was also a descendant of John of Gaunt, it was through one of his daughters, Joan, so he would not have inherited the Plantagenet Y Chromosome that way. The most recent common male ancestor for Richard and John of Gaunt was Edward III, as Richard is a direct descendant along the male line from Edward III's fifth son, Edmund of Langley.
Seniority, on the other hand, determines which child (or sibling) had the right to inherit the throne. Until quite recently, the throne was passed by right of primogeniture, meaning the eldest son (or brother, if the King had no children) became the next King. All sons (or brothers) had precedence over all daughters (or sisters), regardless of their age. This explains why Edward VI became King after Henry VIII died & not Henry's oldest child, Mary. If the King had no sons, then his eldest daughter would become Queen, which is exactly what happened when Edward VI died.
Applying seniority to the case of the Yorks & Lancasters, this means that Richard's family had seniority over the Lancastrians, even though their claim was through Anne Mortimer, who was the great-granddaughter of Lionel of Antwerp, third son of Edward III. In contrast, the Lancastrian Kings (and later, the Beauforts) were descendants of the fourth son of Edward III, John of Gaunt.
I hope this explanation was helpful. I tried to make it as clear as I could, but as always, when you're so related to yourself that you could be your own grandfather, things tend to get complicated & frustrating. But that's what makes genealogy fun!