It has been a sibling rivalry, as well as a commercial one. Adidas
were founded by Adolf Dassler - you may notice that the word Adidas
is derived from the first three letters of his name Adolf "Adi",
and Dassler "Das" - and Puma were founded by Rudolf Dassler.
As you will have noticed, their surnames are the same, due to them
being brothers. The two brothers ran the "Dassler Brothers
Shoe Factory" from the 1920's until the 1940's; after the Second
World War the brothers had a falling out and Rudolf left the company
to found Puma, whereas Adolf renamed their previous company Adidas.
The headquarters and factories of both companies remained in Herzogenaurach,
Germany. The rivalry between the two brothers remained, just as
the rivalry between the companies, it's factories, and it's workers
within Herzogenaurach has remained.