Basically, they're both dirt + water, but silt is very, very fine. You can make a ball of mud, but not a ball of silt.

Silt is granular material of a size somewhere between sand and clay whose mineral origin is quartz ,it is like mud or clay or small rocks deposited by a river or lake. Mud is a mixture of water and some combination of soil, silt, and clay.

Adding on to the above answers, you should know that silt is something that is used for deposits caused by flowing rivers. You must know about THE YELLOW RIVER - Hwang Ho in China. The Yellow River is notable for the large amount of silt it carries—1.6 billion tons annually at the point where it descends from the Loess Plateau. Due to its heavy load of silt, the Yellow River deposits part of its carried burden of soil in its bed in stretches where it is flowing slowly. These deposits elevate the riverbed. Should a flood occur, the river may break out of the bed into the surrounding lower flood plain and adopt a new course. Historically this has occurred about once every hundred years. Thus, sue to SILT, Hwang Ho has caused mass destruction through floods many times in history.

silt; earthy matter, fine sand, or the like carried by moving or running water and deposited as a sediment. mud; wet, soft earth or earthy matter, as on the ground after rain, at the bottom of a pond, or along the banks of a river