One big difference is the AF circuitry, which was optimized to work with the newer MkII supertele lenses to provide faster and more precise AF when using the extender (AF performance with other lenses is the same as the MkII extenders).
According to Bryan at TDP, the 1.4xIII has less barrel distortion and noticeably less CA. The 2xIII is slightly sharper in the mid frame and corners, and has slight barrel distortion (whereas the MkII had slight pincushion distortion - which is generally less noticeable than barrel).
Physically, the 2xIII has two more elements and is 22% heavier; the 1.4xIII also has two more elements (grouped differently, so one less group), but the weight is pretty much the same. The MkIII versions have the fluorine coating on the exposed elements which makes them easier to clean. Also, the MkIII extenders have the 'new' paint color which matches the MkII superteles, if that sort of thing matters to you...