First off, let me get the lulzy comparison out of the way. Yu vs Minato The common misconception is that Yu defeated a shadow that was equal to Nyx in P4. This isn't the case at all. Izanami was not nearly as powerful as Nyx. Izanami was a shadow that represented humanity's want to run from the truth. Nyx is literally death, brought to earth by Erebus, the shadow that represents humanity's want to accelerate the inevitable and die. Now I believe it's pretty obvious why Yu was able to defeat Izanami while Minato wasn't able to defeat Nyx. A shadow can be destroyed however, death is inevitable and cannot be stopped. Later in P3: The Answer we learn that after Minato's death he is actually being used as a seal to keep Nyx from returning to earth. He is forced to defeat Erebus over and over again for eternity because humanity keeps creating the shadow. Tl;Dr Minato is destroying a shadow equal to Izanami in power over and over again, without the help of other persona users, unlike Yu who had to do it with other persona users. Now for the real VS. Tatsuya VS Minato In P2 Nyarlathotep is the main bad guy, who you defeat. Now, this guy could very easily be on par with Nyx. He isn't a shadow. He, like Philemon, is one of the "gods" of the Persona universe. It is unknown if the beings like Nyarlathotep and Philemon are mortal. If they are then Nyx is more powerful than Nyarlathotep, if not then they may be equal in power. Final boss feats leave them equal in power. So let's go on to what happens afterwards. After the party in P3 beats Nyx she goes into "lolnope" mode and knocks everyone down. Minato unlocks his true power and kicks Nyx back into her realm and locks the door with his soul. After the party of P2 beats up Nyarlathotep he goes, "Shit" and destroys the world. The P2 party couldn't stop that from happening. So, reality tears and all that shit happens again this time resulting in Nyarlathotep getting his ass kicked out of the... universe. (Nobody's really sure where he is.) Now that all of this is said and done it turn out that Nyarlathotep is actually coming back (Persona 4 Arena). So Minato permanently dealt with Nyx while Tatsuya just kinda punted Nyarlathotep across the field a ways. Tl;Dr Nyx may or may not be just as powerful as Nyarlathotep, however Minato managed to take care of Nyx for good while Tatsuya did not take care of Nyarlathotep for good.