Not sure where you got that information, did you look at the Dictionary.com site?

Surveillance - a watch kept over a person, group, etc., especially over a suspect, prisoner, or the like:
The suspects were under police surveillance.
continuous observation of a place, person, group, or ongoing activity in order to gather information:
video cameras used for covert surveillance.
See also electronic surveillance.
attentive observation, as to oversee and direct someone or something:
increased surveillance of patients with chronic liver disease.

Spying - verb (used without object), spied, spy·ing.
to observe secretively or furtively with hostile intent (often followed by on or upon ).
to act as a spy; engage in espionage.

As I say subtle but just because word usage can be the same (synonym) there is a difference.