a miraculous turn of events

 הוא אחשורוש – ר׳ יהודה ור׳ נחמיה, חד אמר, אחשורוש – שהרג את אשתו מפני אוהבו, הוא אחשורש – שהרג את אוהבו מפני אשתו,

The statement that Achashveirosh killed his wife [Vashti] because of his friend [Haman] and killed his fried [Haman] because of his wife [Esther] may be a pithy way to capture the irony of the situation, but what does it tell us outside the facts of the story which we already know?  What new insight are Chazal trying to convey?

The entire first two chapters of the megillah seem extraneous to the story.  The main idea is that the bad guy Haman was brought down by Mordechai and Queen Esther, the palace insider closest to the king, who revealed herself to be Jewish.  Who cares what happened to Vashti beforehand, or how/why Esther became queen?  

Alshich and others write that if you read the story in the way I just summarized it, you miss the point.  This is not some story of political intrigue and infighting, where Haman thought he had the King's ear but was upstaged by Esther.  This is not a story of behind the scenes work by "askanim" who brought their political power and pressure to bear to push out an enemy of the Jews.  Yes, of course we can dress that up and say it was not politics alone but yad Hashem behind the actions of all the players, but even so, that's still not the real story here.  Were that indeed the story, then it would have a tragic ending, because there is no way that Mordechai and Esther would not have had the upper hand.  How do I know ?  Because we read in the first two chapters of the megillah that Achashveirosh was so close to his advisor Haman that he was even willing to kill his wife Vashti at Haman's behest!  You can just imagine Haman, after being accused of treachery, falling to his feet and crying to Achashveirosh's, "No, my good lord, banish Peto, banish Bardolph, banish Poins, but for sweet Jack Falstaff, kind Jack Falstaff, true Jack Falstaff, valiant Jack Falstaff, and therefore more valiant being, as he is old Jack Falstaff, banish not him thy Harry’s company, banish not him thy Harry’s company. Banish plump Jack, and banish all the world."  Henry IV Act 2 scene 4 -- just substitute Haman for Jack and Achashveirosh for Harry and the lines are perfect.  Do you think Achashveirosh would take the side of Esther and not only send Haman away, but kill him and thereby "banish all the world?"  Do you think he would do this after already having the sunk cost of having promoted Haman to the highest rank, having killed Vashti because of his advice, having turned over his ring to Haman and given him carte blance to do as he wished?  

The fact that Achashveirosh "killed his wife because of his friend" proves that if Achashveirosh is then willing to turn around and kill that same friend, this time because of his wife, it is a miracle -- not a yad-Hashem-acting-through-the-natural-course-of-events "miracle" in quotation makes,  but a miracle  MIRACLE of the type where something completely unexpected and inexplicable happens.  The first chapters lay the groundwork for just the opposite of what actually happened to happen, if not for Hashem's direct intervention.  This is what Chazal are teaching us by drawing our attention to the first half of the story in contrast to the second half.

What difference does it make in the end?  Who cares if Hashem saved us through the derech ha'teva means of palace politics pushing out Haman or Hashem saved us by causing some supernatural unexpected turn of events? 

The difference is that history does not repeat itself.  The moment in time that saw a king called Achashveirosh on the throne with a queen called Esther and an advisor called Haman will never repeat itself.  There may be similar situations, but what is similar in some respects is different in others.  If Purim was an event that happened through derech ha'teva, then it's meaning is confined to a particular time and place, the parameters that derech ha'teva plays out in.  But if Purim is a supernatural intervention, the who and the when and there where don't really matter.  That same spiritual power that exists l'maalah min ha'teva can be unleashed and does unleash itself time and again.  Teshuasam ha'yisa la'netzah v'tikvasam b'chol dor va'dor.