The debate rages on: one or two spaces after a period?

LawProfBlawg at Above the Law argues in defense of two spaces, linking it to a simpler, easier, pre-smartphone time: “I think the single spacing crowd is comprised of people who are always rushing someplace.”

Mignon Fogarty, Grammar Girl, prefers one space, and notes in Business Insider that most style guides, coding platforms, and typesetters do as well. “Everyone who makes the rules today agrees: It’s a one-space world.”

Another case of old man shouting at kids on his lawn?

Is this discourse simply an excuse for grammatical one-upmanship, or does that one extra space really make a difference?