I do a lot of fire risk assessments and the difference is important. An easy way to see is petrol and diesel, people would assume they are both flammable but only petrol can be described as such. Petrol in its liquid state will not burn it is not possible it is the vapour which burns vapour which is readily given off at most temperatures hence it is flammable. Diesel is combustible as it needs to be produced for want of a better word for it to vaporize in other words it must be heated, throw a match a diesel nothing happens but throw a match at petrol and the vapours will ignite.
Chip pan fires are a good example of a combustible source, the oil once at sufficient temperature will vaporize from the pan to the naked gas flame. Liquids do not burn as there is an absence of oxygen, one of the elements required for ignition it is only once vaporization takes place and it mixes with oxygen you can get a fire. Fuel, heat and oxygen remove any one of these and the fire is extinguished and this is the principle of fire suppression.
Solids such as plastic and wood are all combustible because a process must first take place to release the gases.