As I understand it, both ergo and itaque mean therefore, thus, so, accordingly, etc.
When should one be preferred over the other? Does it depend on context, or do they mean slightly different things?
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Itaque and ergo are both classified as consecutive and conclusive.
A simple distinction would be to state that ergo has a logical meaning, and itaque a consequential meaning in a wider (narratological) context.
It is worthwhile to search for the occurences of itaque in AUC.XXI.
Nunquam ingenium idem ad res diversissimas, parendum atque imparandum, habilius fuit. Itaque haud facile discerneres utrum imperatori an exercitui carior esset. Here you can see the consecutive, conclusive meaning that can be translated as therefore. At the start of chapter 30, itaque is used in a broader context. In chapter 29 is explained that Hannibal's soldier are lacking courage and are fearful. In chapter 30 is described how Hannibal encourages his soldiers. A consequence of the situation that is explained in chapter 29 and introduced by itaque. Ergo can be used in this way in some cases, but we don't see it often.
Until Livy almost all occurences of itaque were used at the beginning of a sentence, it is seldom found in second position and never after a relative pronoun. Therefore you can als conclude that the enclitic -que has not lost its meaning. This supports the idea that is has a function in a wider context, i.e. it introduces a fact that is clear on the basis of the preceding. It can also pick up the main thread of a storyline. The distinction is not always that clear though, because itaque can also have an adverbial meaning, for example: eodem te rediturum dixeras itaque fecisti..
Another argument for a distinct meaning is the use of the combination of itaque ergo in Livy (AUC I.25). This might indicate that those particles belong (or at least belonged) to different categories.
I took most of this from chapter 10 of On Latin Adverbs, by Harm Pinkster.
I'm curious if there is any other recent study available on the particles itaque and ergo. Traditional grammars and dictionaries do not always reach very much depth on the subject of particles, because there was not much narratology involved and it was much harder to do word frequency research in a large corpus of Latin authors without computers and databases.