In the game of cricket, when the bowler delivers the ball to the batsperson, what makes the action of 'bowling' a legitimate delivery as opposed to 'throwing' the ball (which is called a 'no ball')?

There are several ways a "no-ball" can be delivered but the one with that causes the most angst is where the bowler "excessively straightens the arm" when delivering. A movement of less than 15 degrees is permissible. If such a bowler is called it may lead to him/her ultimately not being permitted to bowl again in a match until "remedial action" is undertaken under supervision.
https://www.sportsadda.com/cricket/features/types-of-no-balls-in-cricket/

This explains it.

Throwing, commonly referred to as chucking, is an illegal bowling action in the sport of cricket. This occurs when a bowler straightens the bowling arm when delivering the ball. Throws are not allowed when a bowler bowls to a batsman. If the umpire deems that the ball has been thrown, they will call a no-ball, which means the batsman cannot be given out from that delivery. After biomechanical testing showed that all bowlers flex their extended arms to some degree, rules were changed. Current regulations of the International Cricket Council (ICC) set a limit of 15 degrees of permissible straightening of the elbow joint for all bowlers in international cricket. This law applies between the point at which the bowling arm passes above shoulder height and the point at which the ball is released. The limit is to allow only the natural flexing of the elbow joint which happens during the course of legal delivery.

https://en.m.wikipedia.org/wiki/Throwing_(cricket)