Having purchased Skyrim Legedary Edition (part of the Elder Scrolls Anthology box set), I received the Special Edition automagically in my Steam account on Thursday night. Herewith some quick impressions of the graphic improvements accompanied by comparison images. These were taken on a comparatively old system (Core 2 Quad Q6600 CPU @ 2.4GHz, 4GB DDR2 RAM, Radeon R7 250 w/1GB GDDR5 RAM). SE cuts my frame rate in half from standard Skyrim, but the graphic improvements may be worth it. I especially like the more saturated palette and the water effects.
Standard Skyrim (no mods) is the top image, and Special Edition is the bottom image in each pair. You can click any of the below image pairs for a full-size version. Note the full-size versions are 1440x1800 in PNG format and average a little over 4MB in size each.
Helgen keep, first room if accompanying Hadvar. You can see the darker, richer palette at work here as well as more texture in the moss and rocks. Special Edition has a wider range of lighting as well, contributing to a more realistic look in every area.
Helgen Keep basement just before escaping through the tunnels. Wood and rocks have more texture and SE has darker shadows, but there's less differentiation in this underground area.
Skyrim, just outside the Helgen Keep tunnels. This is the first real "Ooh" moment. The trees and foliage are much, much better looking in SE.
South end of Lake Ilinalta. Another "Ooh" moment. The water is so much nicer in SE and this scene is much more natural-looking.
Overall I'd say SE is definitely worth the upgrade on PC (FREE!). It's also well-worth buying if all you have is vanilla Skyrim without any DLC; OR, you're buying on console and want to play with mods. (Though, to be honest, I'm not sure how nice mods will be on PS4 since Sony won't allow new assets--no new armor!) If you have a newer, nicer PC than mine, SE may pack more of a wow factor. I'm running original Skyrim in HQ and SE in medium quality, so if I could swing HQ in SE it might look even better.