No difference for engine manufacturers between the two terms, the boundary of operating conditions that define where compressor stall occurs is called the "surge line" by all engine manufacturers I have dealt with.
As is often the case, we have several terms for one thing, then define the percieved difference, then get wrapped up.
The compressor stalls or surges when the local angle of attack exceeds the critical value, and believe it or not, the conditions look an awful lot like rotor stall on the main rotor.
The easiest way to get stall/surge is to accelerate the engine too fast, so the N1 speeds up faster than the airflow can adjust. Such acceleration stalls are solved by limiting the N1 acceleration of the engine, and also by relieving the compressor pressure by dumping the compressor discharge air (bleed valves and belly bands).
Other ways to get engines to stall is to disrupt the aerodynamics of the blades (erosion, damage, dirt, ice) or to distort the inflow to the engine (hard to do at helo speeds, but not impossible).
Stalls hurt the engine in several ways: 1) the compressor blades get badly stressed when they see wild angles of attack, and odd make-breaks of the flow, so fatigue damage or even static blade failure are possible 2) the engine temperature in the hot section can get sky high since fuel is flowing but air is stagnating and 3) the cooling air that is passed into the combustor and even the hollow blades is disrupted, so these areas can get hot spots and perhaps even burn through.