In general, "cut" version refers to shortened, or edition with removed/missing content. Example being the Call of Duty Modern Warefare 2 sold in Russia; missing the controversial mission "No Russian" entirely with no option to enable it. Fallout 3 is heavily censored; especially in Germany and in Japan.
The German version is heavily censored; no blood, gore or corpse missing an arm etc. While I am not familiar with the censored German version of the Fallout 3, when I was living in Austria; (German speaking country next to Germany) we bought imported US games rather than buying the German version we knew would be heavily censored. (For PC. 360 have region code, and PS3 won't run any PAL Discs even DVDs; PS1, and PS2 also) Games such as "Soldier of Fortune" had all humans replaced with Androids and the story completly altered from the original which had nothing to do with Science Fiction.
Regards to Fallout 3; I am familiar with the Japanese version. The game does not allow the player to detonate the bomb in Megaton, that part of the quest was actually cut from the original, the room in Tenpenny Tower is simply locked "requires a key" and is labeled "vacant suite" It is impossible to obtain the key and most items associated with the suite. Note: Impossible unless you buy the original as well; in short, defeats the purpose of buying the Japanese version.
As with the German version, body parts cannot be blown off, however only for Ghouls, Humans and dogs. Feral Ghouls, Super Mutants as well as most other creatures including Swamp Folk are not censored/modified. If they actually played the game for couple of hours; they would have learned that, Super Mutants were once human, just like the Ghouls however since gamers and anyone with ties, past, and present with companies in the gaming industry are not eligible to qualify as a reviewer; many of their decisions end up delaying the release date, cancellation of DLCs, and major bugs and flaws rendering so many games unplayable. Most notorious example being Oblivion for the 360. Which Bethesda finally announced recently that they will not be releasing the Shivering Isles. No one cares anymore. (Again, Bethesda is not the one to be blamed.)
example; Bloody Mess perk only adds the extra 5% damage and have no impact on the way enemies die.
As a result; It is impossible to register a headshot during the quest "You Gotta Shoot'em in the Head" despite shooting and killing the target with a bullet to the head, at point blank range. With, or without bloody mess.
example; Unless the player chooses to use an Energy Weapon or a Gatling Laser, all NPC corpses can be dragged and their respawn position moved eliminating the need or desire to enslave them just to make an area free of all hostile NPCs. e.g. Enclave Encampments.
Since players can drag the corpse of respawning NPCs across the entire map, e.g. To force all of them to spawn right in front of Megaton. Which usually results in extremely low FPS, twitching and glitching bodies, weapons being duplicated or vanishing (e.g. Alien Blaster, not just the ammo) Gaining enormous amount of XP simply by planting large quantities of explosives next to the pile of corpses and fast-travelling back and forth every 73 hours. Regardless, since the maximum number of objects that can be displayed and interacted with at once is limited by the game; usually renders the game unplayable.
Interestingly enough, the body of Gary 23 in Operation Anchorage is not missing an arm,(his forearm) thus the holotape describing the "incident" does not make any sense whatsoever. In the US version of Fallout 3 GoTY for PS3, Gary 23's inventory have 2 access points. With the German and Japanese version, it can only be access by the player. In this particular case; it is of no consequence but for instance, retrieving the Holotape from the Drifter requires the player to research before hand, bringing grenades or some type of a weapon that would knock his body down.
The most significant difference is that, release of patches and updates can be delayed by months or years. As for Fallout 3, for the 360, the delay was considerably short. For PS3, updates ended with the fix for the radio bug and support for trophies. They also cancelled the release of DLCs and just recently released the contents on a seperate disc which many claim to be unstable.
One more thing in the Japenese version the fat man was renamed because a bomb that the US dropped on Japan was nicknamed the "Fat Man"