Whenever you study the history of communism, you have to look at it in the same way that a detective would take statements from criminals, and then reconstruct the truth from all the lies, by reading between the lines and seeing through the false alibis.
When it comes to Stalin and Trotsky, reduce the equation to it's lowest common denominator. Power. There was a power play between the two, for the overlordship of the communist world.
Any differences in philosophy were a vast distant second to that primary concern. And in reality, there were no real differences - there was subterfuge.
Trotsky was run out, but he was not killed. His supporters living in the USSR and causing problems for Stalin were dealt with mercilessly, but Trotsky himself was allowed to travel the world blabbing about how Stalin was a bad communist and no legitimate successor to Lenin. Was Stalin too ineffective to deal with him? certainly not. So why? Because Stalin was a master manipulator. He took over a nation with a weak military. The Red Army had shot it's bolt and was in no condition to continue world revolution/conquest. Yet the world still considered the USSR to be a threat. So Stalin manipulated the world by manipulating Trotsky into becoming his greatest propagandist. "Stalin has betrayed the revolution!" (just what the West wanted to hear) "Socialism in one country!" (so the west begins to think that there is no further military threat). Etc. Trotsky ignorantly played the part of the master disinformationist, and Stalin pretended to be angry but impotent to do anything about it.
All during this time, Trotsky was allowed to broadcast his disinformation loudly and without interference. But towards the end, he began to realize that Stalin had not betrayed the Revolution, and was simply rearming, rebuilding, and preparing for round two - along with supporting the rise of Hitler, who would batter the Western world and then let Stalin swoop in as savior and pick up the pieces ( although subsequent events would work out differently). At that point, when Trotsky finally realized the game, Stalin effortlessly had him murdered before he could expose Stalin's true intentions.
And the important thing today is that, for all of his skill which many contemporary historians still don't appreciate, Stalin was a crude child when it came to manipulation compared to the politicians and powerbrokers of today. The people who rule the world against us now study people like Stalin, and go beyond them. And we keep on thinking that pulling the R lever will turn back the clock to 1776.