It's not so much that set is American and that lay is British - rather, that Brits use both forms more or less equally often...

...whereas Americans almost exclusively stick with set...

Personally, I find set slightly more "formal, dated" - but I suppose that's just because half a century ago, my mother always instructed one of us kids to lay the table. We were true peasants, so I always assumed set was a bit la-di-da (obviously it's not).
I think it's a given that since NGrams are drawn from written published sources, they'll tend to over-represent more formal usages. So I suspect that if I could compare formal/informal contexts (contracted I'll lay the table, as opposed to I will lay the table)), this might well support my gut feel about the difference (in BrE - obviously the concept is largely meaningless for AmE). But there simply aren't enough written instances to meaningfully compare.