Velocity helps with longer range shots on targets at unkonwn distance, because if you're off a little bit when you estimate range to target, a faster bullet with a flatter trajectory will not be as far off of your point of aim, and is therefore more forgiving.
One reason guys who hand-load talk about velocity is that for every rifle and bullet combination, there is a certain velocity that just seems to produce the best accuracy, for whatever reason. Hand-loaders often try several different powder charges, producing different velocities, to see which performs the best. Some folks also measure velocity so they can use ballistic charts/calculators to see how much bullet drop they can expect to get at different ranges.
Velocity can also cause a bullet to have greater wounding potential. For instance, the 5.56 military round is a .22 caliber bullet, but it leaves the barrel at well over 3000 feet per second, so it can produce some pretty nasty wounds at close range when it is still travelling fast. Velocity also helps with penetration.
On your question regarding why the .308 is the most common sniper round, I would just advise you not to read too much into which calibers governments and militaries pick for different applications. They make their decisions based on a whole lot of things other than which round actually provides for the best performance. There are a lot of politics, as well as practicalities involving commonality of calibers, etc. involved, and some of their decisions are rather stupid, biased, and short-sighted. For instance, the .308 (or 7.62x51 as it is known to the military) was adopted in the 1950s because the military wanted a caliber that would fit in a shorter action than their old .30-06, but they wanted the same trajectory with the same caliber and a similar weight bullet so the military would not have to change it's training regimen for long-distance marksmanship, and because the top brass was somewhat stuck on .30 caliber, for nostalgic reasons or whatnot. They did this knowing that there were better choices out there... it has been known since the early part of the last century that a bullet between 6.5mm and 7mm traveling at 2800-3200 fps produces an ideal trajectory, ballistic coefficient, recoil, and wounding potential for human-sized targets, and militaries world-wide have tried to adopt such a round multiple times, but it always gets killed because of various political reasons.
Once a government adopts a certain round, they tend to want to keep it, because they already have stockpiles of ammo, it is expensive to buy new firearms in different calibers, and it may involve different training, etc... and then private consumers follow suit, because they like to have a rifle in a common caliber, for reasons of economy. So the situation is that the cheapest and most common rounds are usually not the very most ideal for their intended purpose, but people use them anyway because they are cheap and common. Witness the .223, .308, 7.62x39, etc. etc.
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