First I have to mention that this is first "The Wheel of Time" book written by Brandon Sanderson. Robert Jordan has passed away after writing "Knife of Dreams", so the fate of WoT was in question for a time. Jordan's widow Harriet McDougal picked Brandon Sanderson, at that time relatively unknown writer, to finish the WoT. Jordan's death didn't come sudden, so he was able to prepare lots of notes for what was supposed to be the last WoT book, and even write some parts. Thanks to that, Sanderson had a skeleton of story, and he "only" needed to flesh it.
In "The Gathering Storm", we have two major plot-lines. Rand, after losing his hand by Semirhage in "Knife of Dreams", is continuing his preparation for Tarmon Gai'don. But the stresses, wounds and betrayals he has survived has left a great toll on him. He becomes increasingly harder and darker - and not all Dark One's seduction is obvious. Egwene is still a captive in Tar Valon, but everything points that her efforts are paying off and she is winning the fight for Amyrlin Seat. But what will happen when she is forced to confront Elaida directly - is is better to bend like a willow, or finally make a stand? Nyneve is only major character with important part in this book - as the only Aes Sedai and advisor that doesn't send Rand into fits. Mat and Perrin present, but more with cameos than with real story, and Elayne is completely absent, thus making her the last main character to be absent from one of the books.
First POV in prologue is a great introduction to the book. Until now, we have had a direct insight into fight between light and dark, but always from the highs. But now, this struggle has reached even the common little man. Prologue in general is very good.
This is definitely Rand's book, as it not had been since "Lord of Chaos". He has been under an unimaginable stress for years now, and the time to release it had finally come. I must confess that I never expected Jordan (because I presume that it was his original intention) to make Rand so dark, even evil. There is a curious part of chapter with Rand and Moridin - it is strangely pity for Rand that he can feel comfort near his greatest enemy, who is actually his antithesis. His parts are disturbing, but they are very well written. "I have been forced to revise this particular inclination" - I think this quote gives goosebumps to every fan of WoT. Even though it is not good, I must it is funny to see Cadsuane finally cowed. Book has a very strong ending.
Egwene is only slightly behind Rand in importance in this book. Her parts have been very good for several books now, so there is no need to praise them anymore. She has one of the best action scenes in series, comparable to battle in tFoH or earlier battles (Rand against various Forsaken). Elaida is on pinnacle of disgust here.
As I mention, there is lot of Nyneve, who is left as one of last person with some influence on Rand. She didn't have this much attention since Ebou Dar, if ever. There is also lot of Min, who sometimes look unobtrusive, but she is an important characters. We also have few POVs from Aviendha, which are rare.
Both Mat's and Perrin's parts are not really important in this book - they are "saved" for next one, ToM, and they are here only not to be absent. I don't mind this as such. Perrin's part is actually pretty good, except the small spoiler for next book (meeting between him and Galad). But what I do mind is Mat's POVs. Sanderson blundered with Mat. It is not written badly, but Mat and his humor are very off, and the story is unnecessary and irrelevant. Thankfully, he accepted the criticisms and did much better in ToM.
All in all, Sanderson did a very good job with continuing "The Wheel of Time". There are no major differences in writing, although there are some words and phrases that Jordan didn't use. There is one apparent change: shorter chapters to fasten the pace, with lots if jumps between POVs. My version of the book (Orbit hardcover) has a mistake - Sulin is mentioned as with Rand, except with Perrin. I hope this will be corrected in newer editions.
In general, "The Gathering Storm" is a great book, itself and as part of "The Wheel of Time". Much darker that any before, it really presents struggle, in various forms. And it finally puts in focus the Tarmon Gai'don, the Last Battle.