In my understanding:
1. Fill flash
~ your subject is usually outdoor, and you want to light it up (reduce shadow) by firing a flash onto it. The flash is not the main light, but just to fill in more light.
2. slow sync flash
~ during night time, you want to have your subject light up by flash, but you don't want the background to look dark. In normal sync, your flash will fire at 1/125s, 1/60s, 1/90s...etc (depends on camera), and this will give you dark background. In slow sync flash, your cam will choose the shutter speed as it you are not using a flash at all, and the shutter speed is to allow enough light light to lighten up the background or the whole scene, but yet your flash is to make sure the front subject is bright enough.