The internet does not give a satisfactory explanation as to the slightly differing nuances of these two notions.
When speaking politically, what is meant by "separatists" and "rebels" and where do they differ in ideology?
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Separatists desire to be separate. Rebels desire to no longer be under control.
Rebels are outside of the system, and may or may not want the system to be the same. In Pakistan and Turkey rebels in the army aimed to replace leadership without meaning to fundamentally change the nature of their country. Mao's or Lenin's rebels aimed to remake the whole state.
Separatists but not rebels can work from within or almost within the system. Catalonia recently made a not quite legal bid to be separate from Spain, Texas polls about 10% for independence from the US, Scotland has held legal referendums to leave the UK.
In Sudan and Ireland rebels aimed for a separate state, making them both separatists and rebels.
In the context of independence movements, Separatists are individuals who want to take a landmass out of the administration/control of a country to form a separate country or to join the landmass to another country.
Separatists can be of two types:
Note. Rebellion may exist external to the concept of Separatism. For instance, a group of army soldiers may commit mutiny against the army headquarters irrespective of the cause. That would also be a rebellion. E.g. Bangladesh Rifles revolt.
Note 2. A military coup to overthrow a government is considered as Revolution3, not a rebellion if the entire military is mobilized under the central military command. As a result executors of the coup do not get prosecuted in military courts. For instance, famous coup of Augusto Pinochet was a revolution, not a rebellion.
I would break them down like this
These people want to break away from the government and form a separate government and/or country. This may or may not involve violence. So, for instance, we have
A rebel is not interested in splitting off from the government. A rebel either wants the government replaced, or simply overthrown.
Separatists and Rebels share a common goal but the means to that goal differs between the two. Separatists choose a non-violent way of protesting for their demands. On the other hand, rebels will believe in using weapons against the power they are fighting against. They are often labelled as terrorists by the occupying power. However they are considered Robin Hoods by their own people.