All stoves will burn wood but only Multi-fuel stoves will burn Smokeless Fuel.

Actually, it's not quite so simple:

The best way to burn wood, is by using a top air supply. Air enters the stove at the top, is drawn down the glass (Airwash) and then into the fire. No grate is required nor ashpan. The wood is burned on a bed of ash. True woodburners therefore have only a top air supply, have a solid base and will not burn solid fuel.

Smokeless fuel, on the other hand, requires a ready supply of air up through the bottom grate into the heart of the fire. It is also necessary to drop the ash down through the bars into the ashpan. Multi-fuel stoves, therefore, have both top and bottom air supplies and a bottom grate plus ashpan. Usually they will have a "Riddling grate" which allows the ash to be shaken (riddled) into the ashpan without having to open the door.

But, just to confuse the situation, some stoves termed "Woodburning Stoves, have top and bottom air supplies, a grate, usually fixed, and an ashpan. Whilst termed "woodburning" their instructions will sometimes say that they can burn "selected smokeless fuels". These stoves burn smokeless fuel with varying degrees of success and should be viewed as woodburners with the ability occasionally to burn small amounts of smokeless fuel. The probability is that they have only been tested to EN13240 for wood.