As mentioned, the only thing skins change is the appearence. No stats are affected nor supposed to be. If they were, people would fully classify lol to be a pay2win game, which few does at the moment.
One thing to note though, is that some skins can change the appearence of their spells. I've noticed some times that the abilities are actually harder to spot by the eye on certain skins.
An example is Pulsefire Ezreal, whom in my opinion is harder to track some of his abilities on. His skill shots gets a bit harder to see exactly where the hit area is.
I can aswell imagine Frostfire Annie and her abilities to be a bit more hidden during the winter version of Summoners Rift.
A clear difference is the Nocturne original and Nocturne halloween skin. The Nocturne halloween skin's Shadow ability almost cuts through your eyes due to its strong colors. It might appear to be a bad thing, but can be useful to drag attention with when you're going tanky with him.
But basicly, the only thing skins might improve your gameplay with is visual advantage to a low extent, and the feeling that you must win because, hey, you got a skin !
It seems as if Singed's poison had been purple all along. Never mind this case:
For example Surfer Singed (used to) have an other color on his Poison Trail. It was purple, and it has harder to spot than the normal green one. I noticed enemies running into my purple poison sometimes because it was quite hidden in the brown ground. That is however altered now since they increased the visuals on his poison and made it green for all skins.