Apart from the obvious cosmetic differences and interior spec, when they were both released on 14" wheels the Saxo had Michelin Pilot SX GT 185/55x14s and the 106 had Pirelli P700 185/55x14 tyres. At about T-reg the Saxo VTS went onto 15" wheels with 195/45x15s as standard ( the VTR stayed on the 14s ).
Peugeot did their dampers in-house whereas Citroen bought theirs in from an external supplier. The 106 had a slightly thicker rear ARB at launch ( not sure if the VTS adopted the thicker one partway through its life ) than the VTS.
At launch the Saxo VTS was on Bosch ECU and the 106 on Magneti-Marelli. I think Citroen also used a Sagem one and Magneti-Marelli later.
The early Bosch VTS ECU had a 6800rpm limiter and the early 106 one had a 7000rpm limiter.
I remember spending ages deciding which to get when they were new. They did feel slightly different to drive ......more down to individual cars than any real advantage over one another....and the fact that even the same car will feel different/lap quicker or slower (all else being equal) on different makes of tyres