How something 'plays out' refers to how a situation occurs and develops over time. If you 'play something out' it can mean to complete an activity or see something through to the very end, such as playing a game out to see who the winner is. If something 'plays out' it can also mean to 'run out' or to 'be finished'. How something 'turns out' refers to the outcome or result of something. How something 'turns out' can also mean something that proves to be the case. "She turned out to be right this time." To 'turn out' can also mean to show up for an event or in order to vote in an election. "The turn out for the referendum was 20% higher than normal." These expressions are both phrasal verbs.

1. I'm not yet sure how my afternoon will play out, I suppose it depends on whether Gary can meet me for lunch or not.
2. I couldn't understand why Jane didn't respond to my email but it turns out she never got it. I wish I knew that before I rang her and complained!
3. Over 500 people came. The turn out for the concert was phenomenal.

1. We can discuss arranging another meeting with the client at a later date; first I want to see how this one plays out.
2. Brian had an impressive CV and certainly seemed liked the best candidate, but in the end the job turned out to be too much for him. It was better for both of us when we parted company.
3. Let's play out this scenario and see where it takes us.
4. If we allow this to play out  for a short time in the media, people will forget about it after a while