Mangekyou Sharingan. Same name, quite similar, but still completely different.
One thing I still don't understand is how Itachi uses it. The time he uses MS on Sasuke [in the first flashback we saw], his eyes are the usual sharingan - three commas. Then later, when he uses Tsukuyomi on Kakashi, his eyes are still the same. Come to think of it, we barely ever see Itachi's eyes the way, the Mangekyou is supposed to look like.
Now, when Kakashi uses MS, his eye immediately changes appearance.
Not only that, the same MS they use seems to attack differently. Itachi transfers his opponent to Tsukuyomi, a world where he can control time, space and the mass of objects and tortures him there. Kakashi seems to be able to control the size and position of the barrier space, so he attacks by transferring a part or all of his opponent's body to an another dimension.
Based on those differences, there seems no logical way Kakashi could have actually obtained the true MS. However, we know so little about MS, we could never be sure what
is MS really like and how
does it really work.
So, what can we do?
I have high hopes that Kakashi obtaining a higher level of sharingan has something to do with Obito giving him the eye in the first place. Now that I think about it, the clearest thing we know about being able to use MS is "must kill your best friend" rule. What if Kakashi convinced himself he, kind of, did
kill his best friend? Obito? I'm sure Kakashi feels guilty about the way he died [saving Kakashi], so is it possible that our dear Hatake used that guilt in order to evolve? But then, what did Itachi mean when he announced that Kakashi "couldn't have" done something? Did Kakashi realize the true purpose [whatever that is - another mystery] of mangekyou sharingan by experiencing it?
I have no idea which MS could be more stronger, because Kakashi can't even control his abilities properly yet.
[wow, I'm having too much fun. LOL.]