How can you tell the difference between a second degree and third degree burn?

Pain. 2nd degree burns are very painful because the skin is damaged but the nerves are spared. 3rd degree burns are full thickness skin burns that also destroy the nerve so there is little pain of the actual 3rd degree area.
Yes. First degree burns are basically sun burn second degree burns are typically associated with blister formation. You should at least call your doctor if you sustain such a burn. Third degree burns typically do not hurt much after the initial injury, have decreased sensation, no blisters, and the skin looks waxy and shiny. You need to have a physician see that for proper treatment.