A lot of the below is in the stickys under Metallurgy, or can be found in searches.
In full hardening, the steel is turned into austenite and that is hardened by the quench into martensite. The new brittle martensite is tempered to make tempered martensite. This created a uniform and predictable edge and spine. Generally, this is the best blade.
In differential hardening, the steel is turned into austenite, and the edge is hardened into martensite while the rest converts into pearlite and some mixed structures. This usually leaves a line called a hamon. This creates a hard edge and soft spine, with a visual ( more of less) line of demarcation. The main reason for doing this is cosmetic - the hamon. It is a somewhat unpredictable technique and there may be many failures in doing it. When done right, it can make a stunning hamon.
In an edge quench, sometimes the whole blade is taken to austenite, and sometimes just the edge is heated above non-magnetic. Then only the edge is quenched and converts to martensite. The spine reverts to pearlite. There is usually a considerable amount of auto-tempering of the edge by the heat from the spine bleeding down to the edge after the quench. While some of the "Old Timers" swear by this , edge quenching is very uncontrolled and not nearly as good as full quenching, tempering, and then drawing the spine softer. This leaves a "quench line" which is similar, but not the same as a hamon.
In differential tempering, the steel is fully hardened, and then fully tempered to the edge target (say Rc59). The spine is then drawn to a lower temper with a heat source while the edge is protected from heating any higher than 400F. This creates the most predictable and strongest blade for rough use camp knives and swords.
The long and short of all the above is that tempered martensite is far stronger than pearlite, so it is best to create as much martensite as possible in the initial quench.
In yake-ire, the hamon creation is the desired result of the quenching procedure, so the spine is allowed to become pearlite to make the hamon more visible and dramatic. The blade itself is metallurgically inferior to a fully quenched and properly tempered blade ... but you will get nowhere arguing that with a Japanese smith.
In years past, a blade was usually only Rc 54-55. This made it easy to sharpen, and made breaking a blade uncommon. Today, better metallurgy and better HT processes make a smith able to turn out high quality blades at Rc hardness that would have been considered foolish in the past. A slicer may be as high as Rc 64. Most kitchen and fillet blades are Rc 60-62.
camp and hunting blades are Rc 58-60. Most makers would re-harden a blade that came out at Rc 57. That said, most makers without a good hardness tester ( and using that tester regularly) are likely under-tempering their blades. The hardness should equal the use. Just saying, "My blades are all Rc 61-62" means nothing unless that is the desired use range for the type of blade..
. The process and its benefits fascinated me.
