To "chop" requires a major physical motion that originates at the elbow. Think of someone using a cleaver to cut up carrots or other vegetables. "Cut" refers to a lesser physical motion - it originates the wrist and refers to a motion made with a knife or razor blade or similar sharp device (even the edge of a sheet of paper). "Snip" is a small cut; for example, if you want to open a sealed package of drink mix, you'd "snip" the corner.
However, in your example the phrasing isn't quite right - we wouldn't say "snip off the packet." It would be more common to say "Can you please snip this open for me?"
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