Fans of the original Ninja Gaiden will be pleased with this game, despite some more Western game mechanics, and people who hated rockets in Ninja Gaiden 2.....you should probably look elsewhere for a new hack and slash.
Interestingly enough the story of Ninja Gaiden 3 does actually recognize the existence of the first two, whereas Ninja Gaiden 2 seemed to disregard your blood soaked trek through the Vigoor Empire and back again. Having siad that, you have to unfortunately learn every single move again , otherwise you can't do them. Who would want to start off the game being super powerful and badass. That wouldn't be fun at all.
Ninja Gaiden 3 makes some modern updates to the formula, when it comes to learning and levelling your magic, health, costume, and available skills and weapons. Whereas in the first two games, and their PS3 ports, you had to basically pick up a scroll from a dead ninja or upgrade your weapon to learn anything new, you now get to use all those karma points you're banking up to pay for new abilities. I guess they decided that these meaningless number from before should serve a purpose other than bragging, and I find the change to be for the better.
Golden scarabs also make a come back from the first game, and they are your key to unlocking new weapons and upgrades. Crystal skulls too make a comeback, and have replaced the previous iteration of tests of valor.
All in all, the game seems to be a lot more reminiscent of the first game and it's ludicrous action than the second game ever was. You'll spend a lot of time killing helicopters en masse, blowing up vehicles, fighting actual humans and ninjas and hanging out in real world-like locations, as opposed to the latter chapters of both 1 and 2 where you were in the underworld or an otherwise hellishly morphed location.
The graphics have received an update, and thankfully the story itself actually takes advantage of the new engines abilities to make the game seem halfway serious.
Really the gameplay hasn't changed much at all. Blood fury has been introduced and will probably always be used more than ultimates, but if you're me than you pretty much always use obliteration techniques and swap out some blood fury combos when possible. I don't remember the differences between this game and the original NG3 - but I do remember this game took out some of the terrible mechanics like auto-blocking and the horrible kill cams and such that were in the original 3, in exchange for a more familiar and less easy experience.
There are some incredibly stupid sections of the game, and unfortunately they're attributed directly to the story - rather than for the hell of it - such as fighting a giant dinosaur. Yes, yes, you had to fight a giant skeletal dragon thing in the first game. Yes you had to fight multiple drakes in Ninja Gaiden 2, and a water dragon, and ghost pirahnas. But this, honestly. It's not even the concept, it's the fact that the mechanics force you not to attack it until they want you to.
Ayane, Momiji, and Kasumi were included in the new game as well, because they felt Ninja Gaiden 3 lacked in tits compared to it's previous entries. Fortunately though, with some fan service aside, the kunoichis are actually quite enjoyable to play as, especially Ayane.
All in all, I can honestly say Razor's Edge is the Ninja Gaiden 3 we deserved, and certainly hope the franchise finds new life with this new generation of consoles. If you like NG1 and 2, or Sigma and Sigma 2, please - avoid Ninja Gaiden 3, and get Razor's Edge.
Having established that, the game has some extremely infuriating and arbitrary issues, on purpose, to make it more difficult than it needs to be. One of the most glaring problems is that clearly, someone at Team Ninja played too many fighting games before they worked on this. Elements of Street Fighter and MvC and even Naruto can be seen in this game when one enemy, multiple enemies, or even a boss begins to corner rape you and will literally chain you into an endless string of attacks that both cannot be countered in mid-combo, and instantly kill you.
Interestingly enough, you'll get instantly killed because some idiot also thought you shouldn't be allowed to utilize your full health bar for the vast majority of the game. Ninja Gaiden has always done this neat thing where, as soon as you exit combat, all of the health you lost gets regenerated up until a certain percentage of your health bar. So, this asshat decided "Hey, once you get hit down to a certain percentage, those upgrades to your health that you spent hundreds of thousands of Karma on, they go away permanently until you next save.
This truly wouldn't be a big deal, except NG3 and Razor's Edge both completely eliminated healing items, and items of any sort, from the game altogether. No longer can you pause to look at technique scrolls, switch ninpo or weapons, or use items. Combined with the in-game move list they added, it's pretty clear they wanted this to be more of a pure arcadey action game. As if you had to insert more quarters into your console every time you died and they would profit off of it. Fucking idiots. Nowhere in the game is this more glaring of a problem than the last boss of chapter 4.
As you are often forced to do, you must fight a two phase battle with this boss. The boss's move set doesn't change at all from one phase to another, but your health only restores to a certain point. If you get hit, you're done. You lost. One fury attack from the boss and you're completely screwed. Even if you are lucky enough to have a full ninpo bar, the health it will restore is virtually useless as soon as you get hit once, because there's no way you're charging that entire meter up again with only one enemy who you can only hit sparingly and doesn't die.
If you enjoy self-flagellation and maybe breaking controllers, this is definitely the Ninja Gaiden you are looking for. If you are unfortunate enough to be a completionist and OCD about having to finish a franchise or trilogy, I feel your pain. And I'm sorry.