Military police in English-speaking countries usually only have jurisdiction over military installations and military personnel; they do not perform civilian policing functions except in exceptional situations such as wartime or national emergencies. By contrast, gendarmerie are military police who perform civilian policing functions in normal circumstances.
There isn't necessarily a huge difference in practice between gendarmeries and ordinary civilian police forces; gendarmeries tend to have tougher recruitment standards, a more disciplined ethos, be more heavily armed, their members will have received military basic training which covers skill areas which most civilian police training does not, etc. However, even in countries with gendarmeries, the civilian police often have elite units which are on par with those of the gendarmerie; and, in France, much of the gendarmerie are assigned to policing small towns and rural areas which is often less challenging work than that of the civilian police in the big cities.