Incineration is a general term that applies to burning material, usually waste.  It mostly applied to the plants that were built many years ago that just reduced the waste to an ash with no energy recovery - therefore reducing the volume of waste then sent to landfill.  

Today, with the exception of some hazardous waste treatment plants, all energy from waste thermal treatment plants recover the energy in waste (exported as electricity or heat), and the ash is recycled.  Modern energy from waste plants are therefore no longer incinerators.

The burning process is referred to as combustion, which is the oxidation of the material being burnt in air.  With Gasification, the material being treated is heated to above 700°C without enough oxygen present to enable it to burn.  This causes the molecular structure of the material to break down and produces a gas called synthesis gas.  This enables the gas to be used in various ways.  One way is to burn the gas, and in a gasification energy from waste plant, it is the syngas that is burnt, not the waste itself.  Gasification and incineration are, therefore quite different.