I can think of several good reasons to use American Express, despite it being somewhat less widely accepted. But Discover doesn't seem to have any obvious value over a MasterCard or Visa.
Discover support has been top notch in my experience and after having a number of issues with other credit card providers they've been my goto for a number of years now.
What about American Express? They're doing pretty well. I don't think I've ever seen anyone use Discover, so I don't know how much competition they really provide.
Afaik they're mostly accepted in Europe via their partnership with the payment processing network of Diner's Club (vs. directly seeing a "Discover" sign). Diner's Club is moderately widespread but only in certain market segments, like restaurants and higher-end stores. Not many European ecommerce shops will take Discover or Diner's Club, not even big ones like Amazon [1].
I can think of several good reasons to use American Express, despite it being somewhat less widely accepted. But Discover doesn't seem to have any obvious value over a MasterCard or Visa.