Calling it "special military operation" had an important impact on Russia's internal affairs back in February 2022. General population was reassured that the crisis wouldn't last and that their lives were not in danger, as only professional soldiers were allegedly sent to Ukraine.
This allowed to slowly "boil the frog" until September when the first wave of mobilization started (which, again, was called "partial mobilization"). By that time, people who would have protested against a war in February have already learned the costs of such protests (5+ years in jail) and subconsciously felt the guilt for not protesting earlier and letting crimes like Bucha slaughter happen. At this point, people had motivation to believe it was not a war, so they kept convincing themselves despite obvious sings of the contrary.
Obviously, not calling this invasion a war couldn't fool anyone outside Russia. I don't believe that calling it a war openly would have resulted in harsher sanctions or more businesses fleeing from the country.