Not exactly. Dish does the same that Aereo did - they allow customers to access their own DVR where they recorded information using their own antennae over the Internet. Aereo allowed customers to access their own DVR where they recorded information using their own antennae over the Internet.
Aereo rents out the antenna Dish rents out the satellite dish Aereo rents out the DVR Dish rents out the DVR Aereo allows access over the Internet Dish allows access over the Internet
Dish pays broadcast rights to send things over their satellites to customers' antennae Local TV stations pay broadcast rights to send things to customers' antennae
I don't see a difference.
Take a look at the bold section, notice that it does not say Aereo. Aereo did not pay for broadcasting rights, Dish does, as even stated in your post.
365 Days of drinking Lo-Cal beer. = 1 Lite-year