HOGBOM is the basic CLEAN algorithm. It is robust but slow. CLARK introduces a faster strategy for the search and removal of clean component. However, it can be instable when dirty sidelobes are high. MX cleans the largest region of the dirty map because the source removal happens in the plane. However, it is slower than CLARK because of the repeated imaging step and it shares the possible CLARK instabilities because it uses the same search strategy.
HOGBOM, CLARK and MX may introduce artifacts as parallel stripes in the clean map when dealing with smooth, extended structures. SDI, MRC and MULTI introduce (in principle) a better handling of those extended sources. SDI is a rough attempt while MRC and MULTI introduce the notion of cleaning at different spatial scales.