Hm… not really. I read an article in Japanese that said the two used to be equivalent, and that it’s really just usage that shaped how each eventually came to be used. A Chinese definition I saw said 社 used to refer to an earth god, and subsequently to shrines used to worship it (I think 神社 in Japanese is related), and the Japanese article I mentioned said it used to refer to the ‘village’ god. Basically though, all that seems to imply that 社 has a stronger social (or more accurately societal) connotation than 会. I’d say that in modern Mandarin, 社 usually indicates a more complex organisation than 会, which tends to emphasise the meeting or coming together: banquets and celebrations use 会 (think of 宴会 in Japanese, which also exists in Mandarin). 社, on the other hand, is almost always an organisation of some kind. 会 can be either an organisation or a meeting.
Anyhow, so, here are two possible explanations, both of which are somewhat improbable:
- Perhaps, as I said, 社 carries a stronger nuance of ‘society’ or ‘organisation’, and so it makes more sense to speak of ‘society’ as an ‘organised coming-together of people’ than as a ‘meeting organisation’ (which I think is a pretty good description of a company).
- Perhaps, if shrines were seen as a central aspect of traditional society (I mean, in Ancient China, you were supposed to pray to the earth god before moving your bed or digging holes in the ground), then you could see 社会 as a ‘meeting of shrines’? Or a ‘meeting of organisations’. I mean, society has many levels of hierarchy, whereas I tend to feel like 〜社 designates a single hierarchy or organisation. 会 itself doesn’t imply hierarchy, even if one can often guess how much hierarchy there is based on the specific phrase in which it’s used.
If I had to pick one explanation to stick to, I’d go with the first. Ultimately though, since 社会 is a phrase that was apparently made in Japan, I’d take any well-researched/authoritative Japanese explanation over mine. The trends in 社 and 会 usage do point towards 社会 feeling more natural for ‘society’ than 会社, but it’s really more because 会社 feels more like a single organisation than because 社会 clearly fits.