Let’s make some distinctions and clear up some confusion:
Area scoring: Score is territory + stones on the board (prisoners do not matter directly). Chinese rules use these, but not only the Chinese rules do. For example, Chinese rules have no superko: triple ko is voided under Chinese rules, the same as in Japanese. In almost all internet servers, this is not the case: The “online servers” Chinese rules use Superko, which is not supposed to be the case. That is the case for AGA rules though, which use Area scoring + Superko.
Territory scoring: Score is territory + prisoners. “The Japanese” way. However, there are rulesets which use territory scoring but are not the Japanese rules, and avoid the formal complications in Japanese rules (these rulesets are not commonly used in practice: the Lasker-Maas rules are one example).
There is a common misconception that AGA rules uses “something between area scoring and territory scoring”. AGA uses Area scoring. Under AGA rules, you should fill the dame or you are losing points. Notice that AGA rules require the use of passing stones when counting the Japanese way. When counting the Chinese area score, passing stones are not used. That makes clear that the AGA score is just the normal Area Score, as in chinese rules, but AGA’s passing stones are a nice trick that allows counting in the Japanese style, but getting the same result as by using area scoring. So you can for example play with Chinese rules and count with Japanese style (although nobody usually does that because it is not traditional), as long as you use the passing stones trick mentioned in the AGA rules. But this is just a counting trick: If the AGA rules said “always just count area” instead, the result would always be the exact same result.
Japanese (and/or Korean, since they are the same nowadays) is by far the most common ruleset worldwide. The reason is that Japanese, and not Chinese, were responsible for most of the game spreading throughout the world (think for example of players like Iwamoto, that travelled to teach go).
The formal problem with the Japanese rules is that you have to understand and define in the rules when something is dead. The ruleset is super simple if we all agree to when things are dead, but exactly understanding that is not easy, and depends upon “hypothetical play with no ko-threats, and the special ‘ko-pass’ rule” (this is required to understand why positions like bent-four in the corner are unconditionally dead in japanese rules, no matter if there are unremovable ko threats anywhere else: japanese rules determine life by local perfect play without ko threats).
Of course, explaining “local perfect play without ko threats” to beginners is a bad idea and too complicated (it is complicated to some experienced players too: for example, in this position, http://senseis.xmp.net/?SekiWithEyesQuestion6, the two black stones in the corner are alive under Japanese rules, even though it is impossible to prevent white from capturing and making two eyes: the reason is that in such a case white cannot prevent the placing of a new permanent black stone “in the locality of the group”. This possible but very hard to formalize and explain 100% perfectly). So that is why you don’t really teach the full Japanese rules to beginners, and let them learn the exceptional cases later, if ever: part of the difficulty is that this exceptional cases are really exceptional, so many people play a lot and never see them.
In such special cases, understanding the Japanese rules becomes much more complicated than any other ruleset where disputes are just played out, since life and dead does not have to be defined by the rules in such case. In Chinese or AGA rules for example, if there is any dispute, the rules say “ok, just keep playing: when the players pass again, everything still on the board is alive”. That’s it. The rules stay nice and simple and cover 100% of the cases (although not exactly in the same way that the Japanese tradition dictates in the exceptional cases).
I have compiled clear, complete explanations to the most well knows rulesets here http://elsantodel90.tk/go-rules/go-rules.html (unfortunately, it is all in Spanish for now). There you can see that the Japanese rules contain an explanation far larger than all the rest, which has to deal almost entirely with these “exceptional cases”. Other rules have no exceptional cases, and that makes them so much simple formally. It is not the “usual games stuff”, but mainly the “rare exceptions”, where Japanese tradition dictates differences, and that is what make the rules so complicated to fully formalize (since, if you plan to write the full complete perfect Japanese rules, you must account for every single possible case, not just say “well, let the players see if it is dead” without ever explaining how that is done, which is the approach taken in practice when playing “Japanese rules”, that works fine 99.9% of the time).
in real I preffer japanese though.