The disk method is typically easier when evaluating revolutions around the x-axis, whereas the shell method is easier for revolutions around the y-axis---especially for which the final solid will have a hole in it (hence shell). The disk method is:
The shell method is:
Another main difference is the mentality going into each of these. While the disk method is about stacking disks of varying radii and shape (defined by the revolution of r(x) along the x-axis at each x, the shell method is about vertically layering rings (defined by x, where x is the radius of the ring) of varying thickness and shape f(x).