What is the difference between a change in quantity supplied and a shift in the supply curve?

There are a
few different ways to think about this.

Graphically, a change in quantity
supplied is simply a movement along a given supply curve.  The curve does not move--it is just
that you move from one point to another on that curve.  By contrast, a shift in the supply curve
involves the whole curve moving.

Another way to think about it is to say that
a change in QS is caused by a change in the sale price of the product we are talking about.  If
the price that you can get when you sell fish changes, the QS of fish changes.

A change in the supply curve, by contrast, means that there will be a different amount
of the product offered for sale even though the sale price does not
change.  When the supply curve moves, it means that something else has happened. 
Perhaps, for example, the price of the fuel for fishing boats has dropped.  If that happens,
fishermen will supply more fish even if the sale price of fish does not change.

So a change in QS is a movement along a curve that is caused by a change in the sale
price of the produce.  A change in supply is a shift of the curve that is caused by something
other than the sale price of the product.