Elements isn't "embedded Checkout." Elements is "roll your own Checkout using Stripe's UI components."

It's more than that. The loaded JS library renders your inputs into iFrame elements (for PCI compliance). So it takes that HTML and embeds it down. Recent PCI changes added different-domain hosting for the form itself as a requirement for compliance, which is why you see providers moving to different sorts of embedding.

https://stripe.com/docs/security#validating-pci-compliance

Anyone got any thoughts on when to use Elements vs the new Checkout?

Unless you're a Big Stodgy Corporation that must have exactly your own custom flow, always prefer the Stripe Checkout flow (either new or old) over Elements. Checkout is easier to implement, provides a polished UX, and users are more likely to trust it over your custom flow.

We've got a page for that! https://stripe.com/docs/payments

Use Checkout unless you have a good reason not to. The main benefit is that if there is any large change in regulation (like SCA right now in the EU), Stripe will update Checkout and you will not have to change anything.