Are countersunk screws stronger?

Counter-sinking prevents murder.

Very often, the difference will not matter too much since both screw and wood will withstand 10 times as much as you are planning for anyway.

Do you need to countersink screws?

For soft woods, like pine, a countersink may not be needed, as you can usually just drill a wee bit longer to get the head flush. But for hardwoods, countersinks are a must if you want the screw head flush, or to hide it completely below the surface by placing wood filler or a plug above it.