Veigar is a AP cannon, that much is obvious, but there are some problems that I keep seeing in other builds. Simply, they all rely too heavily on the offensive mastery tree. This at first would seem as an weird statement, but I do not feel that the upper (or rather lower) tier mastery sets in the offensive tree provide a significant trade value for the points necessary to obtain them.
If you have not done so already, consider the mastery guide I have suggested above. Take note of how little there is in the offensive tree, and how much is in the utility tree. This guide is going to provide evidence to show that pound for pound, a utility build is better.

This build is not a full, "here is how to play this guy!". In fact there is a whole chapter devoted to variations on the build. It is, as I hope, a whole new way too look at our favorite yordle.

Veigar is all about ability cooldown reduction, ability power (AP), and ability level. Being able to acquire these elements faster than other champions is crucial for a effective mid game, setting up for dominating the end game. If Veigar is unable to out level the enemy teams mid lane champion, you can easily cost the entire match. To this end, Veigar has Baleful Strike and dark matter. These two abilities allow Veigar to mop an entire mob of minions by himself and reap all the experience therein, along with some AP; however, these come at the price of time.
Time, or cooldown reduction, is the hardest stat to acquire and it is perhaps the most vital for Veigar. It is from this that the whole build is set upon.

The offensive build's best option, executioner, adds 6% extra damage if the target has less than 40% health. This is a powerful option for other champions, but not Veigar. With Veigar, the average AP by the end of the game reaches around 600. Casting primordial burst at that AP will cause 1220 damage. With executioner, that damage goes up a whopping 73.2 damage. Not a whole lot of bang for buck. Not only that, but it is far too situational.
If a squishy (Veigar's bread and butter) has less than 40% health, it wont matter if you have the extra 6% damage, that squishy is dead. If a tank or brute has less than 40% health, 73.2 damage still wont make a huge change. Even if you reached 1000 AP, the damage gained by the 6% will be 102, still not enough to effect a battle unless the enemy is about to die anyway.
I think we can safely say that this option is not for the tiny master of evil.

Next up is the Archmage mastery. Maxed out it provides 5% AP. The effect is overall, meaning that as you gain more AP you get 5% of the new value; in other words, you would receive 30 bonus AP in an average 600 AP match. This is a bit significant, It raises all damage and that is no joke. Yet, in the end you are really spending eight mastery points to get 48 AP (Blast and Archmage together), worth it, but I think that there are more powerful stats to be had in other places.

I have mentioned it before, but it is worth mentioning again. Cooldown reduction is the hardest stat to acquire for Veigar. There are a few items to buy, the most notable being Deathfire Grasp, but they're not all workable for Veigar. You are trading in lots of AP for Cooldown reduction. The masteries on the other hand offer this rare stat, and for the cost of simply 48 AP. With Sorcery and Intelligence you get a total of 10% cooldown reduction. This is an amazing use of mastery points, because it provides a real advantage throughout the whole match.
Early on, it allows Veigar to cast baleful strike that much sooner and gain AP faster. Also, it allows for dark matter and event horizon more often, giving Veigar more survivability, harassment, experience, and over all utility.
To me, this is a no brainer, trading in a little extra damage for what gives me the edge in the whole match.

Not only does the utility tree offer intelligence which just about makes it worth it alone, there are a lot of extra kicks and giggles to add to the fun in this tree.
With Awareness, Veigar gains more experience, leveling faster and being that much ahead of the enemy; Greed you gain more gold, which means better items earlier being that much ahead of the enemy; Transmutation gives a little life steal; there are the mana boosts from Expanded Mind, Runic Affinity, and Meditation; and let's not forget Improved Recall and Summoner's Insight, giving Veigar the opportunity to be back in a lane more quickly.
All of these reduce that frightful period for the tiny master of evil called the lanning phase or early game. It makes it so Veigar can excel more quickly and become the bad*** he really is.

There is a lot that can be varied for this build. Most of the lower masteries can be used as long as you can max Sorcery and Intelligence.
An obvious change would be from Summoner's Wrath and Brute Force over to Mental Force, but it comes down to what a person wants. Mental force gives Veigar an extra 4 AP, added to Doran's Ring gives a total of 19. That AP gives 2.4 added damage to Baleful Strike; whereas, Brute force adds 3 damage to his auto attack, plus with Summoner's Wrath gives 5 AP when ignite is down, which is a great harassment against the enemy champion and out does the buff from Mental Force.
Another variation is to pull points from Greed and put them into Runic Affinity, Summoner's Insight, Good hands, Swiftness, or even Strength of Spirit. All of these are awesome swaps and all have their pros and cons. I have heard that a lot of Veigar players like the Blue Buff, so Runic Affinity might be perfect for you. Beginners may want Good Hands to reduce the time down from death, as all veterans of Veigar can testify WILL happen. Strength of Spirit would be interesting to see implemented, considering Veigar's massive mana supply. Without any items, Veigar would basically double his health regeneration at level 18. Summoner's Insight is great for flashgars or telegars, especially when added to Improved Recall. Just that much less time at the shop.

I Hope that this build guide has given enough facts and has helped my fellow Veigar enthusiasts to look at Veigar in a new light. Please, if you have any insights, constructive criticisms, or concerns comment below.