Hardstyle:

The method that reintroduced kettlebells to the “West”, the hard style, was popularised by Pavel Tsatsouline who created this system with major input by Steve Maxwell. The focus was to take Girevoy Sport and dumb it down in a sense so that the exercise could be used by the general population.

It was a brilliant idea as it really kicked off kettlebell training in the west and it’s usually the first taste of kettlebell training most people do.
Hardstyle wants its participants to exert maximal tension and force when completing lifts to make a person stronger.

Hardstyle uses a different type of kettlebells than Girevoy Sport. Each is a different shape as per the weight and the handles are usually thicker and bigger (I have included a hard style and competition style kettlebell so you can see the difference).
Thus, high rep movements like the snatch are modified so that one completes say 10-15 reps a set instead of using it for a time and doing 40-60 per side for example.

Soft style or Girevoy sport:

Girevoy Sport (GS or) or soft style as hard style advocates call it is a cyclical power endurance sport, where competitors compete in various kettlebell weight lifting events for a 10 minute maximum period. The competitors are divided according to bodyweight and age and weight they are lifting.

Started in the Russian military in the 1940s and becoming a national sport in 1985, the sport has now spread worldwide and competitions are held throughout the world.
The usual events comprise jerk, snatch and clean and jerk (most commonly called long cycle). In jerk, and clean and jerk, males use two kettlebells, while females use one. In snatch, both sexes use a single kettlebell.

A successful rep is counted when the person lockouts a kettlebell, where the arms and legs are extended and the kettlebell are in a controlled and motionless state overhead.