They have a very similar meaning. In normal life, "I didn't" see that coming" is much more commonly used, but either is possible. If you are discussing a film, novel or play which surprises you, then you might well hear "what a twist". A variant. "what a surprise", is quite general in application and more used than "what a twist", I think. Have, a look, if you like, at this frequency graph https://books.google.com/ngrams/graph?content=didn%27t+see+that+coming%2C+what+a+twist%2C+what+a+surprise&year_start=1800&year_end=2008&corpus=15&smoothing=3&share=&direct_url=t1%3B%2Cdid%20not%20see%20that%20coming%3B%2Cc0%3B.t1%3B%2Cwhat%20a%20twist%3B%2Cc0%3B.t1%3B%2Cwhat%20a%20surprise%3B%2Cc0

They are very similar. But "what a twist" is used mostly to describe a story of some kind with a surprising change in the plot. (But also news stories). "I didnt see that coming" is used more generally, and more often re events in real life, and especially things that happen to see or that you personally witness. Eg if your boyfriend surprises you by proposing to you.