Children need to learn how to handle money, preferably starting as soon as they can tell the difference between a one dollar bill and a five dollar bill. (Facility with coins may start earlier.)
Adults have (usually) some form of income that can be categorized as meeting at least basic needs with the rest essentially "disposable."
Children usually don't have access to "earned" income, but they have to learn how to deal with a fixed amount of money to spend, so "pocket" money is good.
Pocket money can help them learn the "value" of money when they have fixed amount of money to spend on what we might call "frivolous" things---in general that helps them learn the concepts of "cash flow" and "budgeting" within a time frame that is meaningful to them at their stage of development.
"Saving" may be difficult for the younger children---so if they want a computer game, rather than having them "save" their money until they have enough to buy it, you might say they have to save up 5, or 10 dollars to buy a game they want with you paying the rest. To make it meaningful (and attainable) make sure the amount that you require they save is appropriate to the amount of "pocket change" you have decided to give them.
And plan on increasing it on a regular basis---they might as well experience inflation.
And perhaps the greatest benefit is the increase in their self esteem from having the feeling of at least some control over their environment.
Cons ?---none if the parents know what they are doing.