I recently watched Curse.NA defeat AL at the ggClassic. In my opinion it was Nyjacky playing Veigar who really made the difference. Once he got to "nuke status" with his DFG and just enough AP he was able to burst down from 100% to 0% any single champ on the other team, or damn near close. The game changed dramatically once Veigar was at his full his full potential as before this it was actually pretty close.
So why don't we see more Veigar play? I know that there are some hard counters to him both in lane and in end-game situations, but I still don't know why for example he couldn't be saved for last pick. Especially for teams like Curse who have such a dominant player like Nyjacky.
Here are some stats from elobuff.com
As you can see Veigar is in the bottom half of played AP champions and is 69th overall amongst all champions.
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Best Answer
Although Veigar is a great burst-nuke damage mid who can counter a bunch of AP champions, with one of the best non-ultimate stuns in the game, and never-ending AP farming with his Q, he does have some flaws that may make him a unappealing pick in competitive play.
His damage comes in a burst which is good for killing a champion, but makes him fairly useless when his skills are on cooldown. He is not like Ryze who can deal consistent damage throughout a whole team fight.
Similar to the first point, his cooldowns are fairly long. His Q being the lowest at 6 seconds, his W at 10 seconds, and his E (The AoE stun) being at 16 seconds when fully leveled.
Veigar can be difficult to learn due to his high skill cap. He has to manage his Q's ability to give him AP for last hitting with it and utilizing it for poke. His E (stun) and his W (dark matter) can be hard to get used to. At level one his stun will stun the opponent for only 1.5 seconds, and his W takes 1.2 seconds to fall. That gives you 0.3 seconds to get your W off after your E before your opponent can dodge it.
Veigar is quite squishy, if the opposing team can initiate on you before you can do your combo you will die very quickly. Veigar has the ability to turn a teamfight into a 4v5, either by bursting down an opponent or by getting bursted himself.
Veiagr also has no ways to escape like other AP assassins. His stun can be hard to aim properly, and can be avoided by walking around it. Also in the early game his stun only last 1.5 seconds, which is not muck time to get away from champions with gap closers.
The best way in my opinion to view this, is that Veigar is an assassin and can either do very well or very poorly. Therefore, it is a gamble to pick him which is why he isn't picked very often in competetive play.