As I understand it, evolution has to have a "need" before a mutation accrues. As with many living fossils, the need is limited, so there is not a noticeable change. Gars and sturgeons likely evolved into a species that required the least amount of need to change.
Because of the issues during the time of the extinction in their period. They had to move to deeper water, so there was a need to change in order to survive. As they used to live in shallow lakes and rivers, but moved to deep sea. But during the same time period, Gars had already evolved in such a fashion, they already were able to survive with the least need to change. Thus the lower molecular changes to their DNA.
That would be my guess, based on my understanding.