The explanation that cleared it up for me was a chapter in Sewell's Logics of History, "2. The Political Unconscious of Social and Cultural History, or, Confessions of a Former Quantitative Historian".

Both were reactions to traditional "names and dates"-style political and military history.

Social history came first. It was primarily focused on what can we know about the little man? It's goal was to write history from below, bottom-up history. Unfortunately, for a lot of time periods, there's just not that much about the people on the bottom. What do you? Its methods primarily focused on grabbing a little information, but on a lot of people. Adde parvum parvo magnus acervus erit. You add a little to a little and you get a great heap. While there were people who did this sort of social history qualitatively (E. P. Thompson, most famously), this work often ended up being quantitative because, quite frankly, there's not a ton of information out there about the little people. You might have a name, an occupation, and an address, and you try to recreate the social world from that. Social history was, especially initially, deeply influenced by socialism and sociology, and so often sought to write the history of forgotten classes. Many of the early social historians did this by looking at things like censuses, cadasters, baptismal records, wills, deeds, slave-keepers journals, etc.

Cultural history was deeply influenced by anthropology, particularly Geertz. People went to gaga for symbols, and meaning. Unfortunately, if you just have a cadaster, you're not going to find a tremendous number of symbols or meaning in it. You can't write a Geertzian "thick description" based on a census record. If you want to get into what people think, that means you probably had to look at different sorts of sources. You get into things like pamphlets, and books, and tracts, and the like. A lot of newspapers. Less labor, more literature. Since for much of history, the poor were illiterate, this often means changing your focus from the workers and peasants to the bourgeoise urban dwellers (not the elite though). Just as social history started as a reaction against the old names and dates history that ignored the working class, cultural history started as a reaction against the social history that too often reduced people to numbers. The question is not merely who these people cared about, but what they believed in.

You can see this in Sewell's own work. One of his first books is Structure and Mobility: The Men and Women of Marseille, 1820-1870. Just from the title, you can tell it's social history. Structure. Mobility. I think he looked a lot at the intergenerational transfer of wealth here, though I forget exactly what his sources were, perhaps census or something. All very quantitive. His next book was A Rhetoric of Bourgeois Revolution: The Abbé Sieyes and "What Is the Third Estate?" Just from the title you can tell there's a shift and this is from his cultural history phase. Rhetoric. Bourgeois. Just one person. That's one of the things about the social and cultural history: many of the revolutionaries of the former became revolutionaries of the latter. They moved from focusing on class to focusing on culture, from historical materialism to mentalités.

Of course, you can have things that are "history from below" social histories qualitatively about just one person, but you often get very deep into what they believed, what they thought, what they hoped for, which is why a lot of the early micro-histories like the Cheese and the Worms are called "cultural history". You can also have social history that takes into account meaning and literary sources. Thompson, among other things, used songs, letters, sports, ballads, legends, and many other diverse qualitative sources in his class Making of the English Working Class.

Today, the two often merge in practice and many people who do social history also do cultural history, and vice verse. Just as its sometimes hard to define a clear break between qualitative sociology and qualitative anthropology, it can be hard to tell the difference between the two today. As both sets of focuses are well-established, and neither seems like an upstart revolution, people feel comfortable to draw on both. You might have a chapter that in the 60's would be thought of as social history followed by a chapter that in the 70's would have been hailed as cultural history. If it's quantitative it's probably "social history", and if it involves a lot of literature, it's probably "cultural history", but these days, my experience is that a lot of people's work falls somewhere between and they tend to think of themselves as both (or they think of themselves as one and others think of them as they other). A lot of works might have a cultural history chapter and a social history chapter, or they might combine the two approaches in the same chapter.