In this particular case, I can't see any difference in meaning between the two example sentences; they both mean, essentially:
All animals are equal, but some animals are more equal than other animals.
No significant meaning is added by including the article the, and no meaning is lost by omitting it.
There are other instances, however, where adding a "the" serves to restrict the language to an implied subset, such as the sentences found in the question linked as a possible duplicate:
He was careful not to say anything that might incriminate others.
He was careful not to say anything that might incriminate the others.
In that case, I think there is a subtle shift in meaning after the article is added.
We could modify the example sentence in this question, and make it to where the the is significant:
Here at the zoo, all animals are equal, but some animals are more equal than the others.
In this case, the the reinforces the idea that we are talking about other animals at the zoo, and not other animals in general.
I'd be hard-pressed to come up with a simple rule that expresses when the restrictive article absolutely needs to be in the sentence, when it should be removed, and when it doesn't matter. It's all in the context, I think – one simply needs to evaluate the sentence to figure out if the meaning changes with the article's inclusion or exclusion.