In an electric heater the coil glows red hot, but the wire running from the plug to it stays cool

Why

Salim's step — more heat, so more current — is exactly the kind of link worth making, and in some situations it holds. Here it cannot, because the lead and the coil sit in one loop, so the same current must pass through both. The difference is how hard each one makes that current fight: the coil is long, thin and made of a metal that resists strongly, so it turns a lot of electrical energy into heat, while the thick copper lead resists very little and stays cool. Picture the same crowd walking through a wide corridor and then a narrow one — same people, but the squeeze is where all the pushing and warmth happens. Test: heater leads are made deliberately thick; if they heated like the coil, the wire in your wall would burn. Divya is right — same current, different resistance.