What is the difference between sparse and dense optical flow?

The short explanation is, sparse techniques only need to process some pixels from the whole image, dense techniques process all the pixels. Dense techniques are slower but can be more accurate, but in my experience Lucas-Kanade accuracy might be enough for real-time applications. An example of a dense optical flow algorithm (the most popular) is Gunner Farneback's Optical Flow.

To get an overview of the flow quality look at the benchmark page e.g. the KITTI or the Middleburry dataset

Sparse optical flow gives you the flow vectors of some "interesting features" within the image.

Dense optical flow attempts to give you the flow all over the image - up to a flow vector per pixel.