Alright let's clear the air a bit here since most of us haven't played Razor's Edge and some of us haven't even played NG3.
First things first, about Ninja Gaiden 3...it's actually a very good action game. It just isn't a very good Ninja Gaiden game. To explain a bit further, Hayabusa is suddenly stricken weaker in many ways by the plot, loses his OT for the most part in favor of Steel on Bone (a similar mechanic but ultimately more difficult to use in combat due to some randomness to it), and there are a few sections where you are rendered completely unable to move properly for story purposes. The kunai climb mechanic is annoying but only happens a few times throughout the game so I largely give it a pass (except on the ice wall...that was nuts). The story is the best in the series, though still nothing crazy, and in my opinion it has some of the most creative boss fights in the series, hampered only by the lack of a health bar so you never really know how you're doing in most of them.
Mechanics-wise, combat flows a bit slower than NG2 and Hayabusa is often forced to fully commit to an attack. You are unable to dodge cancel a much larger portion of your movelist than in previous titles, though shuriken canceling is still as awesome as ever and, if used properly, allows you to cancel some attack animations you couldn't without the shuriken cancel. The relatively annoying guard stun you get stuck in from NG2 also returns, so if you're not careful you can get stuck blocking a full string of attacks with no way to dodge out of it until the attack stops or you're guard broken.
Because you are limited to only the Dragon Sword in the base game (the scythe and talons were later added as free DLC for the campaign) you often don't have as many approaches to combat. Most encounters boiled down to seeing what enemy type you were facing, humanoid or larger monster, and then exploiting something. On higher difficulties this became 100% necessary due to the increased frequency with which enemies grabbed and the sheer damage of the attacks. Lower level enemies will often kill with 1-2 grabs and late-game enemies one shot you all day long. Worse, some enemies don't properly telegraph their throws so it sometimes comes as a surprise from just off-camera. Enemies also tend to shrug off more attacks from you than they react to, so you will end up eating a grab because of the aforementioned issues with being unable to dodge cancel. This leads to many "unfair" deaths since you really couldn't do anything about it.
Also, OL 360 Y/Triangle into Izuna and constant jump-shot use of the bow can get you through Master Ninja in extremely cheesy and unrewarding fashion. I ended up having to resort to it and as a result it was the most unfulfilling experience I've had in a Ninja Gaiden game to date. The gorilla enemies and end-game Alchemist almost necessitated that approach if there was more than one on-screen because of the whole one grab and you're dead situation. Just not fun.
There was a lot to do online, from co-op to adversarial, but it had some netcode issues. In co-op the game would sometimes "split" and you would each be fighting enemies your partner couldn't see until ultimately you had to restart. This really only happened in the later ones which exacerbated the issue. Competitive multiplayer I actually enjoyed, but since most people weren't interested in playing properly and instead ran around mashing attacks out, you either fell victim to loops that were only possible because of latency issues or you steamroll people simply by virtue of understanding the combat system. It was never anything short of all or nothing for me. The battle royale that sometimes popped up (and really the ninja mission system in general) was actually quite a bit of fun and spiced up the proceedings. Ultimately, either due to a skill gap or the lack of variety (again, no other weapons until DLC added them and it was paid DLC for multiplayer), the online died out.
I'll leave my thoughts on the original NG3 there but I could go into more detail if someone asks a direct question.
As for Razor's Edge (and why I'm excited for it), a TON of new content was added over top of the original game. Ayane has several missions interspersed with Hayabusa's main campaign, Hayabusa gets a few of his weapons back (Lunar among them, so thank you Team NINJA), enemies' animations are improved to the point where attacks can be reacted to properly if you're quick enough, and Momiji and Kasumi get added alongside Ayane as playable characters in the multiplayer portions. In single player you now earn Karma as a form of currency to purchase additional moves, attacks, and abilities, adding a layer of customization similar to the older games, and most of the Ninpo from NG2 makes a return, meaning the single Ninpo from the original NG3 is a thing of the past.
The core game still has the same issues I have with it but I'm going to avoid spelling them out here (and I didn't touch on a lot earlier in this post, either). The sheer amount of new content is going to make a lot of it much easier to stomach now, so I'm going to choose to focus on that rather than any issues I had. I am absolutely the kind of guy that gets a kick out of watching Hayabusa leap from the top of a building, dodge rockets fired at him, and impale the poor sap that didn't realize a Super Ninja flying at him is not something to sit and watch. The Ayane missions were a nice change of pace, the slow movement parts with Hayabusa gripping his arm that I briefly mentioned before are removed in favor of some new fights, and the bosses have life bars now, making them much more enjoyable.
There are a ton of previews out there for Razor's Edge, especially the Wii U version, so I won't talk too much about it. I just wanted to put some info out there about WHY I disliked the first version of it. I will say that the treatment by most of the media was very unfair and it's obvious a fair portion of the critics did not finish the game at all, let alone understand what any of the problems were. I saw a few reviews where information was ripped straight from feedback left by users (myself included) on popular forums and articles. This also happened with Sigma 2+ for the Vita most recently, with reviewers not completing the game at all before reviewing, but we can't do much about that. But all of you know me, so I figure my insight will help some of you see things a bit differently.
Tl;dr: NG3 had some major issues, NG3RE doesn't address all of them, but sweetens the overall experience so much that it'll likely be a very entertaining experience despite it all. Buy it and support our favorite developer.