The two expressions essentially mean the same thing. However, play out means how things will develop. Shake out generally implies that some things will be discarded and other things will succeed. For example, you can say we don't know how the elections will play out, meaning what the result will be. Whereas how the election will shake out puts a bit of emphasis on the idea that some candidates will be discarded. In any event both expressions can be used almost interchangeably. Play out is probably the more common of the two expressions. This explanation is from an American point of view and may even change from region to region within the United states.

I would say we don't know how the situation will play out. I've never heard of shake out. Unless it's the action like shake out your clothes for sand. Let's chat more about correct sentence formation. My schedule is open for new enrollments starting today. Please take a look at my profile, reviews and schedule. I offer trial lessons as well. See you this week.