Which swift screen handler approach is better for instantiating a utility class and then presenting a viewController?
The only difference between the first and the second approach is whether 
the handler is assigned to a variable or not. None of them is more 
"functional" than the other. Therefore:


If you need a reference to the handler (perhaps to cancel the request)
then use the first approach.
If you don't need that reference, choose whatever you feel more
comfortable with, or what you find better readable.


Some more remarks:


The method name showWhatNewScreeen is (apart from the typo)
misleading. The method creates a view controller but does not
present it. Something like createWhatsNewController might be 
more appropriate.
According to the API Design Guidelines, variable names should be lowerCamelCase, that is violated at
let WhatsNewVC = ....
The right-hand side of

whatsNewVC.view.backgroundColor = UIColor.black.withAlphaComponent(0.985)


can be abbreviated to .black.withAlphaComponent(0.985), the
type UIColor is then inferred from the context.