Well, most rifles for hunters are churned out by the factory like Fords, Chevys & Dodge & Toyotas, you get the picture.
The sniper rifle is a more precision instrument that calls for greater attention to detail.
For instance, it's not uncommon to find a heavy barrel on a sniper rifle. Our varminters have heavy barrels too but let's put that on the back burner. Hunters don't want that type of weight as they are out for meat and not out to track down human opponents. Hence, the heavy barrel is generally not used by hunters.
Generally both sniper & the hunter's rifle have glass optics. The quality of the optics can vary for both with the deciding factor being the $. While some commercial scopes can rival the sniper scope in $, most of us don't spend that kind of $ on hunting scopes. While it is common today for sniper scopes to have range finding reticles, this feature may also be found on commercial scopes. The difference then is that most sniper quality scopes are design to withstand heavy use in the field. The USMC tested the Unertl by driving tent stakes into the ground with it and mounting it afterwards to see if it would still work. This isn't done with the common Big-5 or other scope for sportsmen.
Base & rings are super heavy duty (and ugly) and heavy on the sniper rifle. Snipers don't care about looks as everything is camouflaged anyway. By comparison, the hunter's rings are rugged but not super heavy. They're also a lot more attractive (can you say Conetrol for rings & base).
Trigger guards. Weight matters. Sniper rifles (or the better ones) have steel trigger guards. They are generally stronger and wont bend as readily and jam up the trigger like an aluminium one. They also allow for more torque to tighten the trigger guard to the receiver. It's generally 65 lbs for the sniper rifle and 45 lbs for the hunter's rifle (including the aluminium trigger guard Rem 700 PSS).
Accuracy - both can be accurate and I have Remington 700 BDLs that will shoot nickel size groups at 100 yards. However, it probably won't do that once the barrel heats up. Mind you, snipers aren't suppose to engage in firefights (their primary mission is to observe and gather intelligence) and if the mission calls for shooting, to be sparse with their ammunition (to reduce exposure to counterfire).
Stock - the hunter's stock can be wood, laminated wood, synthetics. So can the sniper's but the sniper stock must be stable, that is, not affected by temperature and humidity. Thus, whatever climate it goes in, the zero is not affected. Sniping rifles have had wood stocks but these were generally used in the days before the widespread adoption of synthetic stocks.