The Curve website is a little unclear on the USP alright.
It's an app based payment card a little like Revolut - instant notifications, cheaper fx, etc. BUT crucially in normal operation, Curve don't hold any of your money and you never owe them for purchases. Instead the Curve card act as a pointer to or wrapper for your other cards.
You register your actual bank payment card(s) in the Curve app. When you pay with your Curve Mastercard, Curve recharge the payment to your preferred payment card (or your backup card if your preferred card declines e.g. because your bank is having an outage, or they mistakenly tag the transaction as fraud). As the Curve card supports GooglePay, you could therefore use it to wrap PTSB cards.
You can even go back in time and move a Curve purchase to another of your registered cards, after the merchant interaction has already happened (Curve reverse the original payment under the hood - I find this handy if I forget to pay with my corporate card on a business trip, whatever those are!).
It also allows you to change cards/providers without having to update online subscriptions/saved card details - merchants just see the Curve card which never changes (unless it expires or is replaced - but Mastercard now have account updater functionality which most large merchants are now using to transparently pick up renewed/replaced card details).
It's not really a debit card or credit card in the traditional sense, so they don't currently charge stamp duty - at least I assume that's their reasoning.
One disadvantage is that paying with Curve may mean you don't qualify for cashback bonuses from your underlying card issuer.