Does it make a difference in screen reader accessibility to show a success message in the same page a form was submitted? Or is it better to redirect to a different page?
Depends what you mean by "difference".
From a strict WCAG conformance perspective, whether you show a confirmation message on the same page or show on a new page, either is ok as long as you have coded it properly.
If doing it on the same page, use
aria-live="polite" on the container where the message is added so that it'll be read by screen readers automatically.
If you go to a new page, make sure the page <title> is appropriate for the confirmation.
You also might have to think about the "back" button in the browser. If you go back from a separate confirmation page, does that mean you are back on the form and can change values and re-submit it? If you go back from the form that has the confirmation message on it, I assume that would just to back to the page that took you to the form.
So you might have to do a little research and possibly user testing to see which is "better". There might be a "difference" in user experience, but not in accessiblity.