The first thing to note is that all services communicate with one another. So if you call the SAMU saying you need help with a fire then they'll reroute you to the fire department. Now having said this the answer is: it depends on the nature of the emergency. Obviously call the fire department (18) if there's a fire and the police (17) if there's a crime. Regarding health related emergencies the rule of thumb is to call the SAMU when it's a medical emergency, and call the fire department if there's a medical/life-threatening emergency requiring special equipment. For example, you're in a bistrot and someone feels unwell: SAMU. You're walking down the road and you see someone trapped under a heavy object which can't be displaced: fire department (oh and also don't move the object of you might cause septic shock). The reason is simple: SAMU brings ambulances, paramedics and maybe doctors; fire department brings ambulances, paramedics, maybe doctors, and specialised tools.
Now, in some cities firemen are actually running the SAMU. This is the case in Paris and maybe Marseille if I'm not mistaken. The habit of calling 18 rather than 15 for medical emergencies might come from this. It might also be due to the fact that in general the emergencies one might witness on a daily basis often require the specialised action of firemen: fires, vehicle accidents, work-related injuries, drownings, etc.
For English-speaking support 112 is the way to go. Note that 112 interfaces with the local emergency numbers anyways. Also, since English-speaking emergencies add an additional level of complexity to the matter, it's safe to assume that services might take longer to be activated if one goes 112 rather than 15/17/18.
For reference here's a handy list of French emergency numbers and explanations (in French) and even the governmental page on emergency numbers (also in French).