DIOS MIOS replaces the MIOS, which is used for native Gamecube support. This means it can't be used on a Wii-U.
Nintendont can play Gamecube games in Wii mode, meaning it doesn't replace the MIOS. So you can use both.
Compatibility for games can vary between the two, so it's nice to have both. Nintendont is being actively updated, while DIOS MIOS generally is not. DIOS MIOS can use various Nintendo peripherals, such as memory cards and GBA connectivity, and can use virtual memory cards. Nintendont can't use Gamecube peripherals, but I guess it can use other non-Nintendo devices for controllers. Also, it does not support real memory cards.
I'm not sure if Nintendont has an in-game screenshot tool either.