Sapphire cut is the traditional way of cutting natural sapphires. This means the bottom half of the stone is cut with a much 'fatter' belly. The reason for this is to ensure and enhance the color of the sapphire (bigger bottom allows more color saturation).
Diamond cut means the sapphire has been cut with a flatter, thinner belly, the way a diamond is normally cut. The advantage of cutting this way is two fold - it tends to enhance the sparkle of the finished sapphire, and secondly, it allows the sapphire to be mounted in any setting designed to hold a diamond. It also slightly lightens the color of the stone, which can be a positive or negative depending on what shade of that color you prefer.
How to choose which cut?
If you need to put the sapphire in a setting designed for a diamond, then a diamond cut is the obvious choice.
Otherwise, its largely a question of whether you would prefer the color to be slightly lighter but with more sparkle, or slightly darker, as the cutting will affect the color by lightening it (diamond cut) or darkening it (sapphire cut).