If what you are asking is 'What is the universal difference between English with and English and (or their Russian equivalents, for that matter), the question is wrongly formulated, and unanswerable, because other languages need not, and in fact do not, have lexical items with all the same syntactic and semantic properties English with and English and have. Your question should, therefore, as a minimum, be reworded in terms of a hypothetical (?unique!) difference between a would-be universal ?'conjunction function' & and a would-be universal 'comitative/groupal function' C (in the broadest possible sense of 'conjunction' and 'comitative'). The former, at least, is a relatively safe bet, since no human language can work without something (= obviously, not necessarily an and-like word!) that minimally licenses the expression of 'molecular facts/propositions' like [p & q] (where p, q are propositions that name events or states of affairs). Whether a 'group-forming/comitative' function C is also universally necessary is much less obvious, in my view.
However, even that radically abstractive step would not be enough if what motivates your question is the need to understand the situation in Mandarin, where the '?coordinators' we need to render as English and do not really correspond, either in intrinsic meaning or in syntactic category, to English and at all, and where, viceversa, our English ands have to be rendered in Mandarin in terms of completely different things (?prepositions, sentence adverbs, particles, (correlative) verbs, [floating] quantifiers, or even 'zero', i.e.,'pragmatic' devices like sheer juxtaposition, with or without ellipsis for economy's sake, etc.).
The obvious problem with English and (and its counterparts in other IE languages) - as opposed to our hypothetical function & - is that, even when it conjoins clauses, it is often far from being a 'pure' propositional conjunction function. A second, related, problem, of course, is that, at surface level at least, English and (like its nearest IE counterparts) can also 'conjoin' DPs/NPs, and, in general, XPs, X's and even Xs, and one of the consequences is that when it 'conjoins' DPs/NPs, it sometimes (but not in all cases!) is close in meaning to the preposition with (and to the would-be universal 'comitative/group-forming function' C).
As to the former problem, we can approximate English and to a genuinely universal conjunction function & if we disregard the various 'adverbial' (e.g., temporal, causal, contrastive, etc.) implications that and may have and take into consideration only its truth-conditional contribution as a pure propositional function, which we can do by means of a careful choice of examples, cf. below. As to the second issue, of course, the only way to establish the real import of and as a '?coordinator' of NPs/DPs, etc. is to distinguish between 'surface' and 'deep' coordination, which in its turn entails calculating the 'Logical Forms' of the constructions involved in each case.
Consider, for example, the following English cases of 'surface' [NP and NP] coordination and their possible interpretations:
(1) John and Mary died. If we adhere to the abstractive strategy of counting only truth-conditional meaning, (1) has only one interpretation: it claims the existence of two death events with one participant each (= John died, Mary died = Mary died, John died). No particular time sequence or 'groupal/comitative' implication between the two deaths can be legitimately deduced from (1), so and is a pure & function here, in spite of the fact that, at surface level, it conjoins two NPs instead of two clauses.
(2) John and Mary played the flute. Again, (2) has just one interpretation: it claims the existence of 2 flute-playing events (or sets of such events!) with one participant each (= John played the flute, Mary played the flute = Mary played the flute, John played the flute). Again, no time sequence, nor 'groupal/comitative' nuance, is entailed by (2), so and remains a pure & function here, as well, in spite of being a (surface) NP-coordinator.
(3) John and Mary played the piano. In this case, on the contrary, two interpretations are possible. According to interpretation (3a), (3) says that there are two piano-playing events (or sets of such events) with one participant each (= John played the piano, Mary played the piano = Mary played the piano, John played the piano) but nothing follows from (3) as to the order in which each piano playing event occurred. According to interpretation (3b), on the contrary, (3) claims the existence of just one piano-playing event with two participants (John and Mary playing together, four hands).
(4) John and Mary married. Again, (4) has two possible interpretations. According to interpretation (4a), (4) claims the existence of two separate wedding events (with four participants = John plus somebody else different from Mary, and Mary plus somebody else different from John), and, again, nothing can be legitimately inferred from (4) as to the order in which the two weddings occurred. According to interpretation (4b), on the contrary, (4) claims the existence of just one wedding, with John and Mary as the main participants (= John married Mary = Mary married John).
(5) John and Mary argued, finally, has only one interpretation: there has been just one discussion event (or, possibly, a set of repeated discussion events of exactly the same type) with John and Mary as participants (= John argued with Mary = Mary argued with John), not two discussion events/sets of events with one participant each (John or Mary, respectively). In other words, the 'comitative' interpretation is the only one available here.
Now, as far as I know, the Mandarin (?preposition) he2 (alternatively: yu3 or gen1, with some regional variation, apparently) can translate the (broadly) 'groupal/comitative' reading of English and in sentence (5) and in sentences (3) and, possibly, (4) when they have the interpretations (3b) and (4b), respectively, but, to my knowledge, none of them can translate and in sentences (1), (2), or even (3) when the latter has the interpretation (3a), that is, whenever two separate events are involved.
To replicate such English (surface!) [NP and NP] coordination cases, Mandarin must resort to other devices which semantically approximate English and (to the extent it reduces to universal &), but clearly differ from English with - or from our hypothetical universal 'groupal/comitative' function C. In particular, Mandarin can use the 'sentence adverb' ye3 (= also, as well) - assuming ye3 cannot be analysed as an intra-sentential conjunction - or, possibly, you4….you4 (= 'both … and…/not only… but also…', provided the surface 'coordinates' are phrasal, not clausal). All other choices, as far as I know, have non-truth-functional (e.g., temporal, concessive, causal,...etc.) implications (e.g., ránhòu = 'and then', háishi = 'nevertheless', 'still', suo3yi3 = 'so', 'as a consequence', etc.), and, to that extent, they 'say' (and imply) more than English and in the examples above.
Since my knowledge of Mandarin is only elementary, there may well be even gross factual inaccuracies in the discussion above, but they do not worry me too much, as I expect any factual errors of mine will be immediately corrected by the specialists in Mandarin in this forum. That quick, and necessarily selective, discussion was included here only as an example of the way you could yourself proceed to answer your own question in a systematic way: the real point I wished to make above is just that, provided you look not only at 'surface' syntax, but also to 'semantics'/'logical form', English and and with, as well as Mandarin hé, yu3, gen1 and ye3, yòu...yòu, etc., or even a universal & function and a (possibly) universal C function can, indeed, be easily distinguished. Typological distinctions between and-languages and with-languages in cases of [NP & NP] coordination, on the contrary, in spite of all the nice maps, etc., may not suffice for that purpose unless a 'deep' semantic/LF analysis of all the relevant examples has been carried out before positing such language types.
I am no typologist, though, and I do not know whether that has been the case or not, so please do not interpret me as saying anything against typological studies; on the contrary, they are the only way to approach questions like yours, but they must include deep semantic (LF) analysis, not just inspection and comparison of 'surface' syntactic structures.
