Il avait l'air d'être très hostile, certes, mais trouvons-le. Dans l'intervalle, j'aurai su vous convaincre qu'il n'a rien d'une menace.
I understand that the phrase “savoir vous convaincre” literally means “know how to persuade you” and translates naturally into “able to persuade you”.
What I'm wondering is, how does this expression differ in meaning from saying “j'aurai pu vous convaincre”, using the verb “pouvoir” instead?
If you think about the semantic of verbs of possibility, they cover a range of meaning with at one point the (im)possibility being the result of an intrinsic skill of the subject:
I can't drive (because I've never learned how to)
To, at the opposite end, cases where the (im)possibility is wilfully imposed by an outside actor:
I can't drive (because I don't have a permit)
In French, across time and dialects, savoir has been used for the first meaning and pouvoir for the second.
However, there are many cases that fall in the middle of these possibilities: when the (im)possibility is the result of a physical characteristic of the subject:
I can't drive (because my leg is broken)
When the (im)possibility is the result of happenstance, or an accidental exterior cause:
I can't drive(, the streets are flooded)
Or in OP's example, where the impossibility has an ambiguous source: Is it from a lack of skill on the part of the convincer? From the profound stubbornness of the addressee? Is it a unchangeable fact of the universe that a change of mind was impossible?
(The semantics of which verb to choose were already explained in qoba's answer)
Historically, French has used both savoir and pouvoir for such ambiguous sentences, with savoir mostly assigning the cause to the (attempted) doer of the action, and pouvoir mostly to outside or impersonal causes.
In the last two centuries, however, the central dialects have extended the usage of pouvoir to all these ambiguous cases. Belgian French did the exact opposite, extending savoir even to asking a service ("Tu saurais faire X" = "Could you please do x"), which is what sounds the weirdest to speakers of other dialects and is unattested in older texts (1). Speakers from the North, near the Belgian border, and from the Midi are more conservative.
Some languages use a single verb to mark those two types of capabilities. Others use several, but may not split the different meaning in exactly the same way as a given dialect of French. The existence of fuzzy cases where both verbs fit can also cause semantic drift. Consider English, which used to oppose can (ability / internal possibility) and may (permission / external possibility), but where can has almost completely taken over may's functions.
Whatever a learner's native language might do, they should expect some blurring between savoir and pouvoir, especially when reading the French classics.
(1) This extension can be explained if you consider savoir as meaning the cause of the possibility is internal to its subject. Whether someone want to do you a service is after all one of their internal characteristics.
In some regional variations of French, notably Belgian French, "savoir" a le sens de "pouvoir", as in the example you gave. This usage is also more frequent in older texts.
Edit: if you want to split hairs, you could say that "j'aurai su vous convaincre" implicitly refers to the subject's specific skill or ability (persuasiveness, in this case) as the reason for the outcome that is being posited (here, the other person being convinced, whereas "j'aurai pu vous convaincre" doesn't refer to the subject's skill and simply posits the outcome as something that may be achieved. But in practice there isn't such a strong difference and the choice is mostly a matter of personal taste and regional variation in my experience.
« Je pourrai vous convaincre » : j'ai le pouvoir de vous convaincre, d'imposer mes arguments avec facilité, car mon argumentation repose sur la réalité, elle pourrait convaincre tout le monde.
Mes arguments sont beaucoup plus puissants, ont plus de prise sur le réel que ceux de votre hésitation, de votre imagination.
Dans le contexte de la question, il ne peux y avoir de rapport de force entre les personnages, car on ne sait pas si l'hostilité est réelle ou si elle ne provient pas de l'imagination.
« Je saurai vous convaincre » : je trouverai des arguments à opposer aux vôtres dès que vous les aurez exprimés, je saurai vous persuader que vous êtes dans l'erreur, vous démontrer que vos impressions sont des projections de votre imaginaire et non une perception de la réalité.
Dans ce qui ressemble à une discussion amicale, ou en tout cas non hostile, c'est la voie retenue pour amener l'autre à changer de point de vue, car les arguments vont être adaptés à cette personne.
