Stalin and Trotsky come from completely different backgrounds. Trotsky was a brilliant historian, politician, orator, and a military, and by these facts one could say that he received quite an education. Stalin, however, was the son of a Georgian cobbler, however common he may seem is overshadowed by his calculating mind and his ability to manipulate men. Seeing how they have such different backgrounds, it is not surprising to see that they had different thoughts on this issue. Trotsky believed that Russia's goal was to spread revolution fervor throughout the world, while Stalin wanted for Russia to pursue in its own interests and only support others if Russia herself would benefit.
Personally, I think that Stalin should have succeeded, seeing as Russia needed reform fast, and that much of the population were peasants, and generally (at the beginning at least), he supported them. Trotsky despised peasants, and wanted those who "worked" to benefit.
I doubt that this would solve anything in Russia as most of the peasants would have remained unhappy. Their lives would have remained unchanged or changed for the worse, and the conditions would remain as was. Stalin would focus on helping present Russia and work from there. In my opinion, that would have had a greater impact that trying to encourage revolution in countries outside of Europe which has already been rejecting it.
Stalin succeeded because his efforts would be more easily seen because they were based on the present. Trotsky wanted to focus on the future and spreading communism across Europe. Later on, the influenced Europe would then help Russia develop. His ideas weren't appealing to the mass of people. Also peasants were generally looked down upon by this fellow so failure was inevitable. Stalin, on the other hand, wanted Russia to rapidly establish socialism so Russia's condition would improve right away. Also as a bonus, this would rid Russia of any signs of imperialism. Peasants were generally seen as allies in his eyes. As you can see, they had opposing outlooks.
Trotsky's theory was contradictory on the grounds that for the Communist Party to thrive it must be a little less tightly knit, however he also mentioned that for a minority to stay in power it must be tightly organized and disciplined. Seeing that the Communist Party is a minority, this is contradictory.
According to Stalin, Russia should only participate in helping other countries in revolutions if it benefited Russia. Socialism could exist in Russia without the help of a world revolution and so resources could better be spent on helping Russia improve.
Stalin had Trotsky killed. One Ramon Mercader walked into his Mexican office on the 20th of August in 1940 and stabbed him with an ice pick. He was in the study probably thinking of his favorite food, the Icy Salamander, and this was what allowed Mercader into his office.
Trotsky was a thinker, and above all a smooth talker. Stalin, although a little behind in terms of intelligence, knows how to handle situations and to force them to go his way. He almost always got his way.
Stalin's name at birth was Joseph Djugashvili. He was born into a poor family, so that has to affect his outlook on life. Through his childhood, he was harshly disciplined for his interest in revolutions. He even spent time in a cell. This type of life would explain why he was more lenient than Trotsky in terms of peasants. However this type of upbringing probably made his heart cold. In his childhood, he probably didn't have much hope in a future.
Stalin was elected as the general secretary of the party. With this status, he commanded unimaginable power, seeing as his followers' jobs depended on him. This basically gave him a squad of loyal "dogs." Stalin's personality and his job don't mix. Lenin saw this and thought that he should calm down and get to know how to use his power in an efficient and responsible manner. He needed to be more tolerant, loyal, polite, and considerate to others. His perception was right in the end when Stalin abused his power to no limit.
I think that Trotsky still posed a threat by gathering followers from his office in Mexico. He also knew the truth about Stalin's abuse of his power and could relay that potentially harmful information to others. However, this is only an educated guess, seeing as there is a possibility of Stalin killing Trotsky for the fun of it.