Yimmy:
Actually, no. The focuses on training are completely different. A sniper is essentially trained to do just what I described: Sneak in/Remain Hidden, Observe, Identify and Engage(Be it mission-specific target or priority target of opportunity). Almost all snipers double as scouts, even if it's not listed in the job description =) My position is odd, since I'm actually a sniper operating integrated with a platoon, but we're not "ordinary" infantry either, but I had some troubles adjusting(And still have, even after a couple of years).
Meanwhile, the infantry gets all the training required for squad and platoon level operations, operating together with other platoons, often operating together with vehicles etc, and just get the basics of observation training.
I know that the Swedish and British sniper schools put a fair amount of focus on noticing the differences between various weapons being fired, and we even had to take a test on it, determining direction, approximate distance, caliber and if possible what weapons. Orientation of the shooter was also something we should try to notice from the sound(Yes, it is possible, although extremely hard)