Both these words have Greek origin, specifically:
Гиппократ (Hippocrates) comes from ίππος (horse) + κρατώ (I hold) = The one who holds the horses.
Ипподром (horse-race track) comes from ίππος (horse) + δρόμος (road/track).
The word for horse, in original (polytonic) Greek is written as
The aspirate is the one that means an "h" has to be added in English to convey the original Greek sound (therefore "Hippocrates" in English.)
In Russian this (thick) "h" is turned into a "Г" (e.g. Hamburger - Гамбургер).
Why this is not the case with Ипподром?