When burning an open wood fire, the screen's primary purpose is to contain sparks. The best spark protection is provided by either a flat screen or a one-piece bowed screen as either is more effective against spark shooting out over the top of a screen as can happen with a folding screen. This type of screen is also safer around children and pets than a folding screen which is more likely to fall over if leaned against or if a child falls against it.
It may at first seem silly to require a screen in front of a gas log set. However, a screen is required to prevent against a woman's dress or robe from being sucked into the fire due to the drafting of the chimney. If you don't have children or pets in the house then a folding screen will work for this purpose
Freestanding wood stoves or wood burning hearth stoves should have a screen around them to protect little hands and pets from being burned.