Fluorocarbon has basically the same refractive index as water, which means it's a bit harder for the fish to detect. Light passes through it whereas mono tends to reflect light. It is also much more abrasion resistant than standard mono which makes it great for shock tippets and when you might be scraping bottom, rocks or coral. These are the major selling points of fluorocarbon leader and tippet materials.
Fluorocarbon is also more dense than monofilament and tends to sink slightly faster. Although with the short lengths that we use for fly fishing leaders, combined with the fact that you may be attaching a weighted fly and/or split-shot, the difference between the two is minimal and not really a valid selling point. It does break the surface tension easier than mono and many will use it as a dry fly tippet so as to avoid the visible "meniscus" that mono can make, this making for a more stealthy presentation.
Everyone has their favorite...with the Cortland being the best selling, but be sure to compare the value of the Orvis Mirage....it's a good deal.
Our favorite fluorocarbon leader and tippet materials are listed here.