Wikipedia's dating — at least speaking in terms of the widespread adoption of chopsticks — may be off a bit. While they existed as early as the Shang, a better date for their adoption more widely is much later, in the Song (960-1279CE) which is nicely supported by archaeological evidence and changes in human physiology (see Brace, cited below). While they had already moved around the region, their use as eating utensils was more limited. Even more recently they show up in neighbouring cultures but not for eating — a common and in fact sole use for chopsticks in the area around the Burmese/India border was actually for preparing opium and is still practiced today.

Also worth mentioning is that the fork and knife as used today in Western countries is not as old as you might think.

Regardless, In both cases before these points people were eating the old fashioned way, with their hands.

Didn't they have different utensils for cooking and eating?

They did, and still do. The early use of chopsticks did correspond to cooking, but these too were different (and still are) from what you are using to eat – You cannot cook too well in a hot wok with short chopsticks. The most notable difference between cooking utensils and eating utensils, however, is in the cutlery. The Chinese vegetable cleaver (which looks like a cleaver you might see in a Western butcher shop but is much much thinner) is/was a sort of all purpose knife. There are broader and narrower varieties but for the most part it's an all purpose knife. But not one you'll ever see sitting on the dinner table when it's time to eat.

How did the chopstick then spread throughout so many different Asian nations?

The way that they spread is through the cultural influence that China had throughout the larger region. China has historically been the cultural centre and for centuries its neighbours sought to emulate it. It wasn't a question of "our culture and theirs" but rather "culture and less culture" as the prevailing view. Chopsticks spread the same way much of Chinese culture spread through the region – by generally willing adoption as something objectively more developed.

You'll notice that they're found in places where China had great cultural influence like Korea and Vietnam but not in places like Thailand which was outside the Chinese sphere of influence.

Additionally, chopsticks have not always won out in all domains. In Korea, for example, the custom is that rice is eaten with a spoon, not with chopsticks. This possibly developed from the fact that rice often had other grains mixed in, meaning it was less likely to stick together and thus more difficult to eat with chopsticks.

If you're into this sort of thing I highly recommend Consider the Fork: A History of How We Cook and Eat by Bee Wilson which gets exactly to this sort of question about changing food habits. It's a good little read but covers a lot of bases and makes an excellent introduction into the history of modern and pre-modern eating.

• Brace, C. L. (1986) Eggs on the face, ƒ in the mouth, and the overbite. American Anthropologist 88: 695–7

• Wilson, Bee (2012) Consider the Fork: A History of How We Cook and Eat