In baseball, the baserunners are allowed to advance to the next base at any time the ball is "in play" which includes when the pitcher (vaguely similar to bowler in cricket) is holding the ball.
When a baserunner tries to advance without the ball ever being hit, this is called a "steal."
When the baserunner and batter are coordinating so that the batter will try to hit the ball during a steal, this is called a "hit and run." The idea is that the infielders will be getting in position to get the runner out, so won't be in good position to play a ground-ball hit by the batter.
So consider a play where a batter strikes out, then the baserunner is thrown out trying to advance (a strike-out throw-out double play). If it's a swinging strike, that might be a failed hit and run, but if it's a strikeout looking, then it's almost certainly a failed attempt at a steal (with the minority of the time it being the batter missing the signal for a hit and run).
I should acknowledge that 2 strikes with fewer than 2 outs is not considered a good count to try a hit and run; unless a poor batsman is up next you are often better off having a fresh count with the next batter. On the other hand a hit and run is really only effective when it's a surprise.
It doesn't really matter. But it's a datapoint that, say a batter has a lot of strike outs looking, they may have a poor sense of the strike zone. Or if a batter has a high swinging strike out rate then it says that their pitches are deceptive and have a lot of movement.
In baseball you DO differentiate between whether you struck out swinging or looking.
It matters for the pitcher because if you can disguise your pitch well enough that it looks like a ball coming in but is actually a strike, so that the hitter doesn't even try to hit it, that is a great signal.
As a batter, you typically* want to swing at strikes, so you want to know if you are letting hittable pitches go by. From the time baseball players are like 10 years old, you'll hear coaches and parents yelling "be a hitter, don't strike out looking!".
* there are always situations you probably don't want to swing, like if you have 3 balls and no strikes, you usually want to not swing, and assume you'll get another ball and will get walked to get on base. There are a lot of other situations where you do or don't swing, and the strikeout looking vs swinging measures that.