I would say Put the heater on if you want one individual radiator to heat up.
However if you have a central heating system or something that heats the whole house then I would say Put the heating on and that would mean that the radiator in each room will heat up, not just one individual radiator.
I'd usually say "Could you please turn on the heater?", but using 'turn on' instead of 'put on' might just be an American/British English distinction. Also, I think it's technically incorrect to end a sentence with a preposition ('on'), but most English speakers don't pay attention to that rule, especially not when talking.
The distinction between the two words though, is that the heater is the device that provides heat. Heating is the provision of heat. e.g. "This room has inadequate heating; the heater doesn't work very well." However, it might also be short for 'heating system'. This term usually refers to a system, often regulated by a thermostat or other device, that controls several heaters throughout a building. In this case it would be correct to say "Could you turn on the heating?" although "Could... the heater" could still be used. If, however, you're talking about a small space heater with its own on/off switch, 'heater' would be the only correct usage. It wouldn't count as a 'heating system'.
One other thing: Usually, if you're using a space heater, you turn it on if you want heat and off if you don't. This means that you'll say "Could you (or 'I', or 'we' if you want to be really polite) turn on/off the heater?"
However, if you're using a 'heating system', it is usually always on, especially during the winter. In this case you should use 'up/down' instead of 'on/off'.
Other terms you can use when talking about a 'heating system':
In a central heating system the heater can be on but the heating is off - for example in the summer when the heater (boiler) is used only for hot water. In the autumn you have to say "turn the heating on".
But if you have switched the heater off, for example during vacation when nobody is at home, you have to say "turn the heater on".
Yeah but I thought most (like mine) have a "CE" "HW" button, (Central Heating/Hot Water) ... Heating implies Central Heating, nobody would think of "Water", we'd say something like "Hey John, stick the hot water on will ya?"