I understand that the split originated in a power struggle, but I'm more interested in the modern theoretical differences.
First, allow me to point out a subtle category error. Trotskyism falls under the rubric of Marxist-Leninist theory. It's a kind of Marxist-Leninist theory, so asking for the differences between 'Trotskyism' and 'Marxist-Leninism' is like asking for the differences between 'apples' and 'fruit'. Apples may be different from other kinds of fruit, but it doesn't make much sense to ask how apples are different from the category fruit.
All apples are fruit, but not all fruit are apples; all Trotskyists are Marxist-Leninists, but not all Marxist-Leninists are Trotskyists. The categorical inference is not bidirectional.
The main split within the general worldview of Marist-Leninism was between Trotsky and Stalin. Stalin believed that socialists should focus on establishing itself securely within Russia and then expanding outwards, leading to the draconian security state we ultimately knew as the USSR. Trotsky held to the more traditional Marxist ideal that socialists should keep working to foment intellectual proletarian revolutions world-wide, views we might recognize from people like John Lennon, George Orwell, Bertrand Russell, and Bernie Sanders.
Stalin kicked Trotsky out of the Communist party in Russia, drove him into exile, and eventually had him assassinated in Mexico city. But by the time of his death, Trotsky had already formed the Fourth International Movement — in direct opposition to Stalinism — and established his own variant of Marxist-Socialism as a pervasive (if weak and unstructured) presence through much of the world. It would be more accurate to frame this as the difference between 'democratic socialism' (Trotsky) and 'state socialism' (Stalin), but as vast as the difference between those ideologies might be, they are both Marxist-Leninism worldviews.
Trotsky believed that for Marxist revolution to succeed, it had to ignite similar movements globally. Stalin believed contrariwise. But then his main aim was not the emancipation of humanity from under the iron heel of capitalist imperialism but to win the power struggle as to who should be the heir to Marxism-Leninism and lead the Soviet Union. This he did so and then seeing that Trotsky would remain a danger to him had him exiled and then later assassinated in Mexico. It goes without saying that Marxism-Leninism should be distinguished from Stalinism, after all, it was Lenin himself that cautioned that Stalin required watching.
Similar sentiments to Trotsky were expressed by Rosa Luxembourg who then developed a theory which widened the site of conflict from the industrialised countries of the west to the more rural colonial possessions of the west in British India and French Indochina. This is one of the major successes of socialism, the decolonisation movement that swept the globe in the aftermath of the second world war. It's this revolutionary sense of socialism that probably motivated Cuba to intervene in Angola against the racist and capitalist regime of the then aparthied South Africa and prompted Che Guevara to go to Bolivia in an attempt to foment a revolution in Latin America.
Trotsky's view - that for socialism to win would require seeding, nurturing and battling for socialism in many countries - was imitated by the West who then realised to counter Trotsky's vision would require a counter strategy of capitalist imperial containment. Imitation, as Oscar Wilde said, is the sincerest form of flattery that mediocrity pays to greatness.
It's this that's commonly said that motivated the USA's aggressive and genocidal war in Vietnam. However, Ramsey Clark, attorney general during Johnson's administration and then later supreme court justice, went on a visit to Vietnam in 1972 during the height of the war and on the radio station, Voice of Vietnam, angrily condemned the American bombing raids, accusing them of bombing Vietnam "back to the 17th C" and purposefully targeting civilian areas. He later called his own government, echoing Martin Luthor King, "the greatest human rights violater in the world" and that it routinely bullies and mistreats poor, minority and politically powerless peoples. It's this critique that has been taken up post-colonial theorists - after all, both France and Italy had powerful communist parties - and neither were bombed by the US "back to the 17th C."
M-L is normally only used to refer to Maoists. Trots can vary between Orthotrots and new left style non-ortho positions such as the once successful at student recruiting ISO.
In short. Maoists and Trots both agree with the differentia specifica of Leninism: the party as a cross class phenomena which dominates an essentially historically passive proletariat too stupid to know what to do about its own exploitation.
They vary in that each has a specific hypocracy about the form of nomenklatura rule the party pursues. Maoists usually deal with broad tactical compromises with rural proletarians due to political success. Trots make any old shit up to recruit students (see the family tree of Trotskyism as my evidence, https://www.marxists.org/history/etol/trees/ustree.htm ; Callinicos (1990) Trotskysm, http://www.marxists.de/trotism/callinicos/index.htm ).
Western M-Ls without union embedment are just going to be LARPers.
Regarding “any old shit,” consider James P Canons hysteric sectarianism, Posadism’s pro dolphin/alien nuclear war advocacy, or “student organising” by groups concerned with proletarian autogestation who consider students to be non proletarian.