First off, I dispute your assertion that Stalin 'was allowed' to go into power. It is more accurate to say that he schemed his way to power. First, Lenin himself used Stalin as a proxy in a power struggle with Stalin, using Stalin to build support with labor unions that Trotsky opposed, and getting Stalin appointed General Secretary. Then as Lenin grew ill and turned on Stalin, Stalin colluded with Lev Kaminev and Grigory Zinoviev to suppress Lenin's last testament (which was anti-Stalin and recommended removing him from power) until after Stalin was established in power. The Politburo was 7 people at the time. Stalin and allies (three) were opposed to one (Trotsky) and were able to recruit the last three Politburo members to their cause. The main argument was that Trotsky was a revolutionary at the core, while Stalin stood for order. After almost a decade of constant war and civil war, order sounded like a good deal. Also, it may have helped that Trostsky always kind of looked like a maniac, while Handsome Joe Stalin looked like a dude you could trust, or at least that you could cast in a new Netflix historical drama. Stalin took power by making a power play and convincing the other people that mattered to back him. Then he had them all exiled or executed, but that was later.
So that brings us to the big difference of Trotsky. He was all about the revolution, not about dealing with the consequences. From Stalin's ascension to 'person with the most power' in 1924, he was able to exile or remove from the party his enemies by 1927, and start a massive wave of purges in 1934. By pretty much any standard, Stalin moved swiftly and efficiently to consolidate power.
Trotsky, on the other hand, was never that interested in absolute power for its own sake. He was more interested in a collegial rule (i.e. ruling with a group of other intellectuals instead of killing them) and more interested in spreading the revolution elsewhere. This can be seen from his post-exile activities, which never really involved scheming to regain control of Russia, but spent much more time trying to establish united communist groups around the world, and supporting the communist movement in the United States.
So what if Lenin's will had come out, Stalin was ejected from the party, and Trotsky became General Secretary and undisputed champ in 1923?
While Stalin tried to isolate Russia and rebuild it to his liking, Trotsky would have tried to export revolution to Europe. Trotsky spent his exile time travelling Europe and America, absorbing foreign ideas. Stalin was exiled to Siberia. However, even though Stalin was inward looking, he still got invaded by Hitler, and even if Trotsky tried to export the revolution to the Europe, communism never really got a hold there, and he would have infuriated the fascist movements....and he would have gotten invaded by Hitler. Probably not much going on there. On the other hand, if Trotsky really had been trying to openly support communist revolutions in Great Britain and the US, its much more likely that neither Churchill or Roosevelt would have wanted anything to do with him.
So that leads us to the big historical difference. Trotsky's desire to export revolution would have lead the Allies to see him as not much better than Hitler. After Hitler invaded, the Allies would not have been so eager to send tanks and airplanes his way. The Russian resistance would have been weakened, and WWII would have been extended by a few years. In 1945, the Russians would be just turning the tide against the occupying Germans, who would have held Moscow for 4 years at this point. But the Germans really had no hope of winning the war due to another development that occured on August 9, 1945....
Thats right, if Trotsky had won in Russia, Truman would have had no choice but to nuke Hitler into unconditional surrender. Maybe Stalin was the lesser of two evils....