Why don't we see more Veigar in League of Legends?
Solution 1:
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.
Solution 2:
Veigar has a lot of issues throughout the game, requiring careful management and calculation.
As has been mentioned - Extremely squishy, requires proper building, easily killed by aggressive bruisers if the team gets distracted in team fights.
I think the biggest issue against him, is that people view him as an AP nuker (don't get me wrong, he is). So even players on his team assume he's going to nuke one target down, and be useless. However, with cooldown, a skilled Veigar will utilize the AOE stun every 11 seconds. That's potentially a full team stun every 11 seconds, which is absolutely brutal. Once the Veigar breaks away from going for the kill and transitions to a Team Fight controlling aggressor, you hit a whole new level of play style.
A team built around Veigar should include a tank with a snare/stun (to peel aggressors), solid adc, tanky side of support, and AD bruiser third position. To take full advantage of his long range catch and dominate capabilities, and as every fed Veigar results in a transition to heavy MRR countering, the AD focus for the team will punish any attempt to mitigate his damage.
Also, Veigar provides very easy opportunities for early game ganking (pre-lvl 6). Strong CC, moderate damage, if he's controlling his mana supply, you can easily force the mid lane into poor positioning and early deaths (or at least force them out of lane prior to level 6).
Why is he underplayed? Any mis-management of skills, mana, etc will be miserably punished in today's meta, making it a very unfriendly learning curve these days.
-Guntrik