Have you considered protease digestion and SDS-PAGE? As the conformation
of the protein changes, protease sites become accesible or are buried in
the protein. This leads to changes in the pattern of bands on the gel. If
you want to push it, you can then isolate relevant bands and do
N-terminal sequencing, to find out which parts of the protein are
involved. Trypsin and chymotrypsin are frequently used for this purpose.
Very gross changes in protein conformation (like oligomerization) can be
analysed by physical techniques like gel filtration, low angle light
scattering, ultracentrifugation, native electrophoresis.
Of course, the holy grail in such studies is to crystallize the protein
in both conformations and get the structures with x-rays.