As I understand it, PR-API is a glue for current payment methods. So your browser would simply contain a hash that is connected to a payment method (i.e. a credit card). It's effectively like Stripe built in to your browser. Before you use it you need to put in a payment method.

GNU Taler, however, is a payment method. It federates with other financial institutions to allow you to make online purchases without using credit cards. In the SHA2017 demo, they keep saying "it's like cash" and that's very close to accurate. The one difference is that payment recipients are forced to keep a paper trail of what money they've received (not from whom, but just that it was received). So in that sense it's almost better than cash in that it's more transparent for tax purposes, but maintains the anonymity of cash for the buyer.

Once the Payment Handler API [1] is completed and implemented in browsers and browser extensions can register themselves as payment handlers, GNU Taler could be used through the Payment Request API.

[1] https://www.w3.org/TR/payment-handler/