Checkout works through the API: you create a Checkout Session specifying what's being sold, the price, and where to send the customer afterward, then redirect them to a Stripe-hosted page that handles the card form, available wallets, and tax calculation on its own. It fits a business that already has a website and a backend but still wants Stripe to own the payment form itself rather than building one from scratch.
A Payment Link is the no-code version of the same idea. You build it once in the Stripe dashboard, get back one shareable URL, and send it by email, text, a social post, anywhere, without writing any code at all. It's a good fit for a one-off sale or an invoice sent by text that doesn't justify a full Checkout integration.
Link, despite the similar name, has nothing to do with Payment Links. It's Stripe's own saved-payment-method network: once a customer has used Link anywhere that accepts it, Stripe can offer their saved card at your checkout too, which usually means a faster, higher-converting checkout for a returning shopper.