Update 10-5-2016: Bethesda announced mod support will be coming to PS4.
Bethesda announced in a blog post today that user-created mods for the Special Edition of Skyrim will not be coming to the PlayStation 4. They claim that Sony has refused to allow mods "the way they should work," but did not go into much detail:
"After months of discussion with Sony, we regret to say that while we have long been ready to offer mod support on PlayStation 4, Sony has informed us they will not approve user mods the way they should work: where users can do anything they want for either Fallout 4 or Skyrim Special Edition."
For more information, read the full update here: https://bethesda.net/en/article/3i0mUzrfp6ioMU6WWcWmKm/ps4-mod-update