Most Dankort in Denmark are combined with Visa as Visa/Dankort or with Mastercard as Mastercard Dankort. As a consumer, you do not experience that there's a difference whether it is the Dankort part of the card that you pay with or not. However, there is a difference.
Paying with Dankort there isn't a spending limit on how much money you can spend on your card each month. Therefore, you do not risk your payment being rejected and your card blocked, even if you have money on your account.