Earlier quoted context omitted.
To be more precise: Vercel is the company (fka "Zeit"). They have a suite of complementary software products and services, including NextJS (a "batteries included" React framework with robust support for SSR and SSG), Now (hosting / CDN / runtime in "serverless" paradigm), Micro, etc. World-class DX and remarkably powerful tools that somehow combine simplicity and flexibility. No affiliation, just a longtime grateful…
Vercel is the company and the product. Now is the old name of the product. It's been thoroughly re-branded. If you go to now.sh it redirects to a page on vercel.com with no mention of "now". The command-line tool is "vercel", aliased to "vc", and it is configured in vercel.json (no longer now.json). If you go to GitHub and look at the repos containing with "now" in their name, they're archived or deprecated.
FWIW I liked "Zeit" better than "Vercel" (shrug). But preferences aside, I was annoyed that the new redirects for "now.sh" URLs were -- uncharacteristically for a team famous for attention to detail -- mishandled, e.g. breaking a fantastic preso I'd bookmarked (which grauch used in his React Conf 2017 talks) and which I had cause to reference today in promoting NextJS with a client. It's the only real "footgun" move I've seen them make. Here's hoping it's an anomaly!