I'm entirely aware of what a soldier trained as a sniper does. What I'm saying is that Bullgrit's distinction between the two doesn't hold a lot of water since sniper is more of picking a method of doing what you're doing than the difference between, say, soldier and assassin. The sets do intersect (sniper and assassin) and can do so easily.
And you may not see a difference between picking someone off at 100 yards vs 500 yards, but I assure you, the unit the sniper targets does. Woe to the sniper who gets caught.
The biggest difference is that a sniper operates in the theatre of war during a time of war while an assassin is not constrained by either restriction.
If you are in a unit in hostile territory (and that could include a town your troops are occupying) during a war, a sniper may very well attempt to pick off key personnel. The thing is,
war has been declared and, according to a famous adage, "all's fair" - including sending someone to pick you off at a distance (no matter how much you may not like it or what you'll do to that @$#%^@< sniper if you catch him). You are inherently at risk from hostile soldiers and you know it.
With an assassin, someone - a religious order, a foreign government or just someone who has a major problem with you - wants you dead, war may not have been officially declared (though it may have been declared by one or both sides) and you may well be in a "safe" area - your home area or neutral territory. You are not expecting imminent hostilities - though you may be aware in general that you have powerful enemies who may well want you dead.
The assassin may opt to use a rifle like a sniper does or may opt for something closer, quieter - the method is unimportant, what is different is that you are not specifically expecting to be attacked at that time or any time in the foreseeable future.
It's "WTF? I'm in Switzerland, I should be safe here." or "I'm feeling kinda funny after eating my breakfast - oh, crap!"