【Abolish】 , 【annihilate】 and 【extinguish】 all mean to make nonexistent.
【Abolish】 implies a putting to an end chiefly of things that are the outgrowth of law, customs and conditions of existence.
【Annihilate】 suggests a complete wiping out of existence of something material or immaterial. (see also: 【Annihilate】 vs Decimate )
【Extinguish】 is like to suggest a complete but gradual ending (as by stifling, choking, or smothering). (see also: Crush vs Quell vs 【Extinguish】 vs Suppress vs Quench vs Quash )