I fully support "I Blame Communism"'s idea that political traditionalism does not equal economic backwardness. In fact, if 20th century is any indication, it is the other way around. USSR, Asian Tigers, China made significant industrial progress while ruled by some most dictatorial regimes known to humankind. So, longer life of AIII does not necessarily mean decreasing tempo of industrialization comparing to OTL. In fact, it was AIII who started serious industrialization in 1880s and laid down Russo-French alliance (after Germany allowed Reinsurance Treaty to lapse), basically defining NII's European policy.
So, what would longer reign of AIII spell for Russia? 1st thing to spring to mind is increasing level of social and ethnic (as far as policy toward Slavic minorities and Trans-Caucasians within the Empire was concerned, Finns and Central Asians were less affected, Jews were actively pushed to emigrate, although rapid raise of Jewish population, caused by improvements in health care, defied the policy) unrest. AIII's way to deal with social ills was to gloss them over and to increase police repressions against dissident groups. Taking into account that pressure within the boiler was rapidly raising OTL (Empire was shaken in 1905-1907 and erupted in 1917), repressive policies might have an opposing effect, causing violent revolution around 1905. It is far from certain, but pretty possible.
It is possible that longer reign of AIII would delay WWI or butterfly it away altogether. French IOTL were sure that Nicholas II would come to their help in the case of war, so they steered pretty recklessly in their German policy. And Serbs pretty much steered Russian court to whatever direction they wanted ("tail wagging the dog"). AIII (although he, through influence of his Danish wafe was pretty anti-German) would be less reliable ally and less prone to dangerous moves in support of Serbian radicals. Again, just like with internal situation within Russian Empire, inherent instability of European political system post-1870 is not going to disappear, but more cautious policy of Russian monarch could lead to flashpoints like Saraevo being handled in more diplomatic fashion (a mere uttering from SPb "we don't approve" would probably butterfly OTL marriage between Black Hand and Serbian government away).
And last but not least - American (and probably Israeli) Jewish community today would be much larger ITTL. AIII's policy (as described by Pobedonostsev, who was in driver seat there) was somewhat along the lines "keep 1/3 of them behind the Pale, assimilate 1/3 of them, push the rest into immigration". So, would AIII rule longer, more Jews would leave Russian Empire.