In swordsmithing, the essential difference was that east Asian swordsmiths used a refined process of folding layers of different metal/metal-compounds together into a single blade. In Japan, the largest source of iron was in iron sand, a product of erosion of larger rocks. Being sand, it was a lot easier to extract the iron than if it were found in rock, however the percentage of iron yield was very small and hard to manipulate.

To add on to this problem, the technique of Japanese samurai was specific that a blade had to both be sharp enough to cut but be strong enough not to break (the West had a similar method, though not to the extent of Japanese swordsmanship). With the quality of metal found in the mountains, they soon found out that sharp was brittle, and strong was dull, so to stop this they layered many (several to a hundred) metal alloys over each other (hence folding). In Japan, most swords were composed of a soft iron core between two hard iron bars, whereas in China, it was more common to literally insert a separate piece of steel (generally with higher carbon concentration) into a folded 'casing' of layered steel. The higher the number of layers, the better the quality of the blade.

Both Chinese and Japanese swordsmiths, however, used the same (possibly with small variations) technique of differential heat treating. Originating in China, a swordsmith would pack the sides of the blade with refractory clay (essentially clay that retains strength at high temperatures) while leaving the edge exposed. The blade is then heated and quenched - science tells us that this rapid decrease in temperature crytallises the carbon edge while leaving the rest of the blade strong and not brittle. This method is featured more prominently (and made famous) by Japanese katanas, as shown here and here.

The difference then, between east Asian swordsmithing and European swordsmithing was that the European smiths had access to better quality steel to start with, allowing them to simply shape, hammer, and grind to fit. East Asian smiths essentially had these four techniques: forge-wielding (combining multiple layers of metal), cross-section rigidity enforcement, differential heat treatment, and repetition of forging to increase the quality.

I could go into more detail but I'm kind of swamped in a lot of stuff so if you have further questions please ask.

Sources:

Technology and the culture of progress in Meiji Japan - Wittner, D.

A History of Metallography - Smith, C.

Could you specify whether you mean "blacksmith" as a general metal worker, or do you mean "blacksmith" in the more technical sense: an iron worker? (i.e. a bladesmith is a related but different trade to a blacksmith)

How about the other way round? Would a Japanese blacksmith know how to work his way around a European forge?