In very simple terms.
Stall is what happens when the compressor blades stall, similar to an aeroplane wing, often caused by ingesting something.
Surge is what happens immediately after a severe stall condition, the high pressure air downstream of the compressor surges forwards. In an axial flow type engine the compressor is normally destroyed as the air surges forwards, some of the resulting debris will then be reingested and generally destroy the hot section of the engine. I have seen this happen at close quarters on a military jet running at high power at night. Quite spectacular, loud bang, sheets of flame coming from both intake and exhaust, followed by silence disturbed only by the noise of aircrew running away.