I don't have either of the cards.
We've seen categories overlap occasionally. I don't think it happens enough that it should stop you from having both cards if they both appeal to you. It simply gives you a second option.
I have an AMEX card, and like Discover, its acceptance is less than Mastercard and Visa. That's no big deal if you're armed with a Mastercard or Visa when you encounter those situations. AMEX and Discover cardholders expect the card to not be accepted from time to time.