Propane provides more energy per unit volume than does natural gas. Heat is measured in BTUs, or British Thermal Units. Propane provides about 2500 BTUs for the same volume of natural gas that only gives 1000 BTUs.
Since they are both petroleum distillates, made of hydrocarbons, these gases have comparable power. You probably couldn't tell the difference between their performance on a grill. They ignite at the same temperature.
Some barbecue enthusiasts prefer natural gas because it is a cleaner burning fuel, which means it doesn't release as many pollutants into the air. Exhaust from propane might affect the taste of food. To the end user, there aren't many differences between natural gas and propane, but they do differ in their chemical properties.