Latin has at least two words that straightforwardly translate to English "know":
Plautus combines the two pleonastically:
nec vos qui homines sitis novi nec scio
Here's one way of asking my question: Are there any meanings that are not shared between these two words? If so, can you give examples of cases where one would be appropriate but not the other?
This is quite a large question, to which a comprehensive answer is easily found in the more comprehensive dictionaries under Know, with copious examples. However, in direct answer, there certainly are meanings not shared between the two, which I will try to summarise, although it is worth pointing out that the differences in usage can be very slight.
Scio is the most general word, meaning that you have a certainty, or at least clearly perceive, some fact(s) or other. It is followed by the kinds of clause that you would expect: acc. + inf., de + abl., relative + subj., neuter pronoun, and so on. Its opposite nescio, (‘not to know’, ‘be unaware/ignorant of’ etc.) is used similarly. But scio (and nescio) can also have the sense of ‘know how to …’ (particularly where it refers to a skill) as in scio scribere, nescio aratro uti.
The simple idea behind ‘not to know’ is expressed by ignoro (this being possibly more definite than nescio, which can be qualified by, for example, a clause after quin). The opppsite of ignoro is nosco, meaning ‘am acquainted with’, which is more usually seen in the perfect tenses, still giving a present sense in English. To claim a personal acquaintance, say, you might appropriately introduce yourself with te novi, or me no(ve)ris.
With, I think, a shade of meaning rather more active than that of simply learning (for which disco/didici or certior fieri is appropriate), the verb for the sense of ‘getting to know’ or ‘finding out’ is cognosco, most often in the perfect tenses to imply the knowledge for which scio in the present might sometimes be used equally well. Comperio is different again, with the sense of ‘know for certain’ or ‘tried and tested by experience’; compertum habet = ‘he knows without doubt’.
This is only an educated guess of a partial answer, but in line with Tom's more general answer, I think.
Disclaimer: My knowledge of the issue is not deep nor technical, and could be biased by the Spanish verbs conocer and saber which are to some extent valid translations of nosco and scio respectively.1 Please let me know if I am mistaken.
I would use novi (not scio) to mean to have met someone.
Notus is the participle of nosco and in a specific sense means acquaintance, friend (at least post-Classically, L&S only lists the plural noti in this sense). It seems to me that there is no parallel with scio for this meaning.
From the definitions, it seems that one can scire and noscere something, but only noscere someone.
I can find examples of novi+person, but none of scio+person without some deed by that person being the thing that is actually known.
Example:
Novi enim te, novi patrem, novi domum nomenque vestrum (Cic. Lig. 5) / for I know you, I know your father, I know your birth and your name
1 I'm well aware that conocer is derived from co-gnoscere and saber from sapere, but that is not my point.
