So I often hear these terms in game. I am not sure what the difference between global skill and global cooldown. Can someone explain if there is any different or they are the same?

They are really different things:

Global skill is a champion skill that effects the whole map "globally". Examples like Soraka's ultimate which effects all allied champions, no matter how far they are, or Shen's ulitmate, who can teleport to any allied champion on the map.

Global cooldown is a cooldown on your skills that affects all other champion skills. So if you use Sona's Q for example, all other 3 of her abilities will be on cooldown for a few sec.

udyr and sona are the two main champs that involve global cooldowns, basicly when sona uses one of her moves all her other moves enter a global cooldown state of 2 seconds, its basicly to keep her from just throwing move after move out, its a seperate cooldown then her normal one, Udyr has the same thing, to keep him from stance dancing much to quickly, global cooldown is uneffected by cooldown reduction.

basicly global cooldown means a cooldown that affects all of one champs moves, and is seperate from each individual cooldown.

"global' range skills can hit anyone regardless of range.

Also, all champions have a very short (probably less than a half second) global cooldown on all their abilities.

List of Global Skills (They are referred to as Global Ults also):

Annie: Tibber can technically go anywhere in the map.
Ashe: Shoots a giant ice crystal the flies in a straight line until it hits a champion
Ezreal: Giant arch of light stuff that can hit multiple targets, flies in a straight line.
Gangplank: Cannon barrage in an AoE anywhere on the map.
Heimerdinger: While only a half global skill it is considered one to me. He can cause his turrets to shoot slowing bullets and he can be anywhere on the map when he uses it. So it's a half global skill, much like Annie's
Karhus: Damages all enemy champions everywhere.
Pantheon: Flies into the sky an lands causing a massive AoE slow effect.
Shen: Teleports to an ally giving a protective shield.
Soraka: Instantly heals all teammates.
Teemo: Similar to Annie and Heimer in only being a half global skill. Teemo sets down mushrooms constantly, when hit they cause slow and poison.
Twisted Fate: Reveals all enemies on the map and teleports anywhere he pleases.

I hope I got them all.

testing

Twisted Fate and Zilean also have global passive skills: Twisted Fate increases all gold earned from kills and Zilean increases all experience earned.