Sets a floor on the spread: a market cannot be tighter than one tick.
The consequence when the tick binds. If the economically fair spread is narrower than one tick, competition cannot express itself through price, so it moves to speed - everyone quotes the same price and the winner is whoever gets there first. Much of the latency arms race exists because ticks are coarse relative to fair spreads in liquid instruments.
Tick size also affects queue position value: coarser ticks mean longer queues and more valuable places in them.