I see redundancy here. If one stands a car on the street, isn't it necessary to stop first? It would seem the most logical and efficient use of language for the sign would be "no stopping." Or do drivers somehow differentiate between the two acts to the extent that they need to be told to avoid both?
No Standing means you can stop to let people out of the car or pick up someone who is already there, but no waiting for someone or loading & unloading. Remember it as ‘people only’.
No Stopping is the most restrictive. You can’t stop at the curb for any reason you can think of. This one is pretty much ‘don’t do it’, or ‘think before you stop’.
On the TV show Parking Wars this question is brought up frequently. On one epsisode they said that standing is used because of cities that have designated taxi stands where the taxis line up waiting for customers. They explained that in that context cars cannot stand (stop) at places with those signs for any reason or length of time. They said it has nothing to do with pedestrians standing.
I recall a similar discussion here in Sweden. IIRC the reason was along the lines of Peter Shor's comments. It is possible for a car to come to a stop without the the driver stopping it in the normal sense, e.g. the car breaks down, avoiding an accident etc. In this case it was not possible to give it a ticket for stopping. That's why the standing part was added, although at first sight it seems implied. (In Sweden the sign doesn't have any text but translated the sign means "Prohibited to stop and park vehicles")
It is definitely implied. "Standing" is extended stopping.
"Stopping", in this context, means being stopped while operating a car, even briefly, for a reason other than to avoid conflict with the flow of traffic. Note that "stopping" does not mean bringing a car to a stop, but operating a car while it is at a stop. So once the car is stopped, the driver is still stopping so long as the car remains stopped.
"Standing" is extended stopping, with the car remaining stopped for more than a brief period of time. So if a sign says "no standing", you can still stop briefly, for example to load and unload passengers.
"No stopping" signs necessarily prohibit standing as well. All the time you are standing, you are also stopping. "No stopping or standing" or even "No stopping, standing, or parking" is to avoid confusion among those drivers who, inexplicably, haven't read or understood their State driver's manual which quite likely explains these concepts clearly.
Note that "stoping" is a form of mining. So if you see a "no stoping" sign, be sure not to remove any ore from the vicinity of the sign. For example, if you see a woman wearing this shirt, you know you cannot remove any ore from her -- at least, not that evening.