So my wife had made me go see the Broadway musical Wicked and the new Disney film Oz. Can anyone explain to me which story line is correct when looking at the Wicked Witch of the West?
That depends on your definition of "correct".
If you assume that the original L. Frank Baum novels are canon, then both Wicked and Oz, the Great and Powerful are both "wrong". Though the 1939 movie has many issues with respect to the novel, the way she is portrayed in the movie is actually reasonably close to the book (unlike Dorothy, Glinda and the Good Witch of the North).
There have been quite a few stories set in Oz that are completely non-canon, many of these are entertaining in and of themselves. (I recommend Tin Man) Though they may be good stories, that does not and will never make them part of the official Oz canon any more than Hardware Wars or Spaceballs are part of the Star Wars canon.
In the cases of both Wicked and the Disney film, there are many examples of characters, situations and plot lines that were introduced by the creators of those works that are clearly identifiable.
For example, we know from the Baum books that the land of Oz was at one point ruled by four wicked witches. Two of them (in the north and south) were deposed by good witches who then assumed control of those lands. This one point alone is an important part of Oz canon history as much that follows stems from it. It also clearly reveals both Wicked and the Disney movie as completely non-canon.
The only story that is "correct" is Baum's book.
Everything else is to varying degrees imaginings of other people which may have something or nothing to do with the original book.
Wicked is what is known as "apocryphal" story (which Wiki defines as "The adjective apocryphal is commonly used in modern English to refer to any text or story considered to be of dubious veracity or authority, although it may contain some moral truth. In this broader metaphorical sense, the word suggests a claim that is in the nature of folklore, factoid or urban legend"). In other words, it's not "true" or "real" story from the point of view of the main universe. It clearly contradicts Baum's book, by design.
Based on Wikipedia plot, Disney's 2013 Oz is not exactly an absolute faithful follower of Baum canon, to put it mildly, but it is clearly NOT in the same league as Wicked in reversing who's good and who's bad.
One reviewer characterized that movie in a way that I found very convincing, by comparing its relationship to Wizard of Oz to that between "Star Wars" and those fanfic pretend prequels about clones and whiny kids. It's an origin story that is in many ways NOT in tune with the original Universe, but is retconned enough to pass as truth... from a certain point of view :)
Justin Chang of Variety had a mixed reaction... He also compared the film's scale with the Star Wars prequel trilogy adding, "In a real sense, Oz the Great and Powerful has a certain kinship with George Lucas' Star Wars prequels, in the way it presents a beautiful but borderline-sterile digital update of a world that was richer, purer and a lot more fun in lower-tech form. Here, too, the actors often look artificially superimposed against their CG backdrops, though the intensity of the fakery generates its own visual fascination.