They are both 'wrong'. Use stresses at integration points (which are by definition unaveraged). Nodal stresses (averaged or unaveraged) are interpolated from integration points and should only be used to make pretty pictures.
If the difference is large, it means your mesh is not refined enough.
i think unaveraged stresses means the raw results. this means that you can expect to see four different stresses at a node shared by four quads. the reason is different element stiffness for different element geometry or different applied loads (the cg of the elements are at different locations in the model = different internal stresses).
averaged stresses then have only one value at each node, averaging the available element stresses.
averaged results are probably reasonable if there isn't much difference between the individual elements.
possibly the better thing to plot is element centroidal stress, which is an average of the element stresses.
The purpose if performing a grid-convergence study is to ensure that these quantities are essentially the same. If they are not, then your mesh is too coarse.
