A multi-fuel stove can burn wood as well as solid fuels such as coal, however you cannot burn coal on a dedicated wood burning stove.
Wood burns best on a bed of ash and burns from the top downwards, because of this dedicated wood burning stoves do not require (although some have) a grate or firebars which allow for air to reach the fuel from the underside.
Grates can be of the fixed or riddling variety, fixed (as the name suggests) are immovable. Riddling (movable) grates or firebars allow for the fuel to be ‘riddled’ which is the term used for the removal of ash from the combustion chamber, this also serves to ‘stoke’ the fire. Most stoves with a riddling facility allow this happen without having to open the stove doors.
Coal burns best with combustion air fed from both the bottom and the top of the fuel, for this reason coal burning stoves or multi fuel stoves are equipped with grates or firebars. Another feature of a multi fuel stove is an ashpan. The ashpan is the metal pan that sits in the bottom of the stove collecting the ash that falls through the grate, by riddling the stove you cause ash to fall through the firebars/grate into the ashpan. This allows for relatively clean removal of ash from the stove. It is important that you do not allow large amounts of ash to collect in the pan before emptying; ash has pretty good insulating (reflective) properties and doing this can cause extremely high temperatures directly under the grate which can lead to warped, cracked or even completely burnt out fire grates and bars.
Generally multifuel stoves cost between 5-15% more than there wood burning equivalents (where a manufacturer offers both options for a model of stove) but the extra cost is usually worth it as it gives the owner the flexibility of choice of different fuels and the practicality of an ashpan for easy cleaning.
If you live in a smoke control area choosing a multifuel stove (that can burn approved smokeless fuels) as opposed to a DEFRA approved wood burner means that the range of stoves that you can look at is not reduced.