If you've ever used mods on your Skyrim save, it probably won't work on the Special Edition.
At this point, I think that most everyone playing The Elder Scrolls V: Skyrim on PC has used at least one mod. It's hard to blame you, as there are mods that do everything from add new hairstyles to add lots of new content. But there's some bad news out there today: any Skyrim saves that use mods probably won't work in the Special Edition.
According to the documentation provided to outlets with review copies of the Skyrim Special Edition (emphasis mine):
"Existing save games from the original PC game will work in the PC version of Skyrim Special Edition. Simply copy your old saves from My Games/Skyrim to My Games/Skyrim Special Edition. Only use original saves that have never been used with mods. Do not use your original save game if this error appears when you load it in Skyrim Special Edition: 'This save relies on content that is no longer present.'"
Apparently the Skyrim Special Edition and Skyrim are considered two separate games, rather than just two versions of the same game. That means that most mods will likely need to be updated to work with the Special Edition as well. It also wouldn't surprise me to see a mod come along that makes these mod-contaminated saves compatible with the Special Edition.
The Skyrim Special Edition launches today on PC, PS4, and Xbox One.