I think some of those suggesting the use of nails are confusing what happens with a house that is protected from the elements to wood that is left out in the weather to get wet, dry out and move around. Nails are the worse thing possible for holding wood together that is exposed to Mother Nature. Ring shank nails have their place, and will hold a little longer then screws, but nowhere near as well as screws. It's night and day.
Neither nails or screws are to be used for structural loads, so comparing sheer strength of either is moot. Both will fail if used improperly. Both are designed to hold lumber in place, not support any kind of a load. Imagine if there was no nail or screw to hold it in place, would that piece of wood stay there on it's own for a second without the fastener? If yes, then a nail or screw is good. If not, then a structural fastener designed for that application is needed. Thing like joist hangers or bolts or even structural screws are then needed for that specific application.
If the engineers that designed the play set say to use screws, then you should use screws.