The Tiger was the result of the German Armaments Ministry issuing a specification for a new heavy tank to Daimler-Benz, Henschel, MAN and Porsche in 1937. The initial design contract was awarded to Henschel. In 1939 Porsche was also contracted for prototypes. However, the "Tigerprogram" was not given much priority until 1941 when the shortcomings of the Pz III and Pz IV were becoming apparent.
I don't think anyone's suggested that the Tiger II was "a follow on" to the Panther; certainly not I. The Panther was a MAN design and the Tiger II a later Henschel design intended to share as many components with the Panther as possible, including engine and suspension components. Some Panther components were redesigned to facilitate this too. The Tiger II, as the name suggests, is an evolution of the Tiger I, with thicker sloped front armor and a higher velocity 88 mm, the revised "breakthrough" tank, even if not the one best used to further exploit that breakthrough and actually, fast becoming the tank more useful to stop a breakthrough than start one. It was also, very importantly, easier to manufacture than the original Tiger.