Silt and clay are both the result of the physical and chemical breakdown of the minerals in rocks. They main difference is in chemical composition and particle size. Silt is composed of silicate minerals, or those containing silicon and oxygen. Clay is composed of metal silicates, or silicates with metals like magnesium or aluminum associated with it.
In terms of handling the two as part of an experiment, the main discernible difference is particle size. Sand particles are larger than silt particles which are in turn larger than clay particles. Silt particles are about 0.05 to 0.002 millimeters in size. Clay particles are smaller than micrometer in size. So silt particles are too small to really be felt with your fingers like grains of sand. But wet silt will feel very smooth like finely ground baking flour. Clay particles are so small that they will simply feel sticky. You won't be able to move them around your fingers like silt particles. This stickiness will be the main difference between the two in terms of touch.
Another experiment to tell silt and clay apart from each other is to agitate the soil in a clear plastic bottle filled with water. The soil is dispersed throughout the water and it appears very cloudy. Any sand particles will settle to the bottom very quickly. Silt will settle to the bottom over the course of a couple of hours, leaving lightly cloudy water remaining. The cloudiness is clay particles which will take a few days to settle on top of the silt. You can then see the layers as they form in the bottle.