The main UX differences between pages and dialogs are:
A dialog can have fixed dimensions, whereas a page has to scale. I'd suggest sketching the screens in your payment process for both a tablet and a smartphone in this case.
In a person's mind, a page is a new place, whereas a dialog is the same place. Consider how the page looks and whether it's clear what it belongs to. If a person switches to another app and then switches back, will it be immediately clear what the page is for?
Dialogs can be transient. It's not as disorienting when a dialog disappears as when a page disappears.
Personally, I'd recommend going with a modal. It works on a number of screen sizes out of the box and it's not disorienting. Android Pay uses a dialog, Stripe uses a dialog, and you can be sure that they've done some user testing to make sure that it works well.
That said, it doesn't necessarily mean a page is the wrong way to go. If you want to be thorough, make a prototype with a page and with a dialog and test it. Or do an AB test in your app.