Difference between Cut Edge and Deckle Edge Paper

Paper is made out of the fibres, mostly of wood or cloth. These fibres are beaten to a pulp, and put in a bath with water.

Now, the deckle is the (wooden) frame that is used to control the size of the paper:

Mold and deckle (source)

This deckle lies on top of the mold, and these are used to scoop pulp out of the bath.
As the water drains through the mesh (the netting on the top of the mold), the pulp will stay inside the frame of the deckle, and the fibres in the pulp will settle. They will not form a straight edge, but a wavy and irregular, or 'feathered' edge: this is called the 'deckle edge'.
They can be seen as the original or natural edge. Often, however, paper sheets are cut to remove these deckle edges, and these are called 'cut edges'.

The difference between the two is almost purely aesthetic: the deckle edge gives a more authentic or original look to paper. But it also is often a sign that a paper was handmade: higher quality papers used for the arts often have deckle edges to separate them from cheaper machine-made paper (since the machines used for paper-making produce paper on rolls, so no deckles are used).