熱中症 I think is the more general term, for pretty much any kind of hyperthermia regardless of how it is caused and also pretty much regardless of severity.
While 熱射病 in particular refers to a severe hyperthermia caused by excessive heat from the environment that would require immediate medical intervention.
I also tried googling and it mostly overlaps with my intuition, but it also seems like people don't always use the terms in the same way. This way of separating these heat disorders seems to be quite broadly accepted though: https://www.otsuka.co.jp/health-and-illness/heat-disorders/type/
In normal everyday life if you're not speaking to medical personnel, I would always just use 熱中症 and maybe explain the details if they matter, but I don't think the other terms would be correctly understood by everyone.
熱射病 or 熱中症
Which one of these translates into heat exhaustion? Jisho defines both as heatstroke.
Merriam-Webster distinguishes the two as follows.
Heatstroke - a life-threatening condition marked especially by cessation of sweating, extremely high body temperature, and collapse that results from prolonged exposure to high-temperature
Heat exhaustion - a condition marked by weakness, nausea, dizziness, and profuse sweating that results from physical exertion in a hot environment