"During Napoleon’s winter campaign of 1812, the Russians and the French experienced the same freezing cold. However, it wasn’t the low temperatures that destroyed Napoleon’s Grande Armee as it is often described in various sources...."
https://www.rbth.com/history/334546-no-winter-didnt-destroy-napoleons-army-general-frost
One can certainly not say that winter alone caused the destruction of the Grande Armee. However, one also cannot dismiss the climate's impact on the end of the campaign. In my opinion the main cause of the 1812 defeat was Napoleon's stubborn refusal to admit the Tsar would not treat with him which culminated in the late departure from Moscow with inadequate preparations for the march back.
Having set the stage for a less than glorious trek home with bad decisions and leadership the effort was made worse by 2 critical considerations; the weather and the fact that the action undertaken was a retreat.
The Russian winter was worse than any experienced by the forces that made up the French Army. Both the horses and the men suffered all through the effort. Possibly worse at the end, but complaints of the cold were heard at the start as well as the finish.
I disagree that both sides felt the same discomfort to the winter. The Russians had experienced the same level of cold/snow/ice before. They knew what to expect and were mentally, physically and logistically prepared for it. Not so for Napoleon's force.
The other major factor contributing to the depth of the defeat was that the act itself was a retreat. Retreats are also something not shared equally between the opponents. Both cover the same ground in approximately the same time, but the one retreating does so usually with lower morale and a propensity toward more bad decisions, lost equipment and medical issues.
The Russian winter alone did not doom Napoleon's 1812 force, but it was certainly a contributing factor which helped turn retirement from the scene into a military disaster.