The gas that is used as fuel in an outdoor grill is propane. When stored in a tank it is actually a liquid but that is only because it is under high pressure. At atmospheric pressure and room temperature it is a gas. It is a hydrocarbon composed of three carbon atoms and the associated hydrogens. The structure is below:
Rubbing alcohol is 2-propanol (also commonly called isopropanol or isopropyl alcohol). It is a liquid at atmospheric pressure and room temperature. Its structure is based on propane but it has a hydroxyl group (OH) on the center carbon:
The only difference between the two chemicals is the hydroxyl group. The remarkable thing about the hydroxyl group is that it has the ability to form hydrogen bonds with itself in an intermolecular fashion. So the hydrogen atom on one hydroxyl group makes a hydrogen bond with the oxygen atom on another hydroxyl group on an adjacent molecule. This hydrogen bonding network has a stabilizing effect. Without it (like in propane), the boiling point is so low that it is a gas at room temperature. With it (like in 2-propanol), the boiling point is raised so that it is a liquid at room temperature.
So the reason that one is a liquid and the other is a gas is because of the hydrogen bonding involved with the hydroxyl group.