Kingdom of Amalur > Skyrim
Starting with what Skyrim does better.
Immersion: Skyrim
Skyrim has a more interesting story and occasionally I stopped to read a book. In KoA I basically spacebar-ed through the dialog.to get to the action.
World: Skyrim
Skyrim has more varied landscape than KoA, and in KoA you can't jump, which I think lowers the "fun" feeling of exploration.
Now what KoA does better.
Combat:
Not even close. KoA is so incredibly awesome in fights it makes Skyrim look like tetris. I had very little strategy or variation in Skyrim fights. In KoA my strategies vary significantly because of all the possible move combinations. Part of this is the ability to quickly roll away, or if you choose a mage path like me, you teleport the same distance instead. This leads to many awesome move/weapon combinations.
KoA is like a button mash game, only if the mashing was very deliberate. It's very fast paced and rewarding.
Weapons: KoA
Ties in with combat. In KoA, I actually have 4 different main weapons that I switch in between frequently during combat. And the type of weapon varies tremendously. I use daggers, circular boomerang things (Chakrams), staffs, and staves (like fast long range staffs). Each has their own combos that work well in certain situations.
For instance, I would use my chakrams to fight slow melee enemies, and as soon as they got close I would use an area of effect combo. Then I would follow up by switching to daggers and attacking one opponent. If another enemy got close, I could blink away very fast because daggers are inherently fast and allow it. I couldn't blink away fast with my staff because each attack takes a long time, but my staff does a better area of damage. So each weapon has it's own place in a fight.
In Skyrim, I used one standard weapon the whole time until I found a better one.
Enemies: KoA
Skyrim enemies were very similar in nature. KoA doesn't have as many variations, but the variations it does have actually vary. Another cool thing is that time slows when a troll slams his hammer on or close to you, and speeds up afterwards. It sounds like a bad game mechanic, but it actually makes combat a lot more exciting.
Loot: KoA
KoA loot is just better. The loot you find could actually be better than the stuff you're using, and it's not like Skyrim where you spend time figuring out what loot to keep so you can sell it. Also KoA has a "junk" menu, so you can put the stuff you plan on selling out of the weapons menu, significantly reducing clutter.
Leveling: KoA
In Skyrim, each level hardly does anything. It takes a a while before you actually notice a significant difference. In KoA, each level adds a skill or upgrade. Leveling is more rewarding. Also, choosing your upgrades in multiple paths opens up different possibilities that you wouldn't otherwise get. The upgrades synergize with each other, something that doesn't really happen in Skyrim. You have more choices in Skyrim, but they aren't as awesome as in KoA.
TL;DR
Skyrim isn't as fun as KoA.