Pumice and cinder are both volcanic rock. They form in a similar way and they look a lot alike. However, they are quite different in a number of ways, as well.
Pumice is a rock with a high content of silica, which tends to make it light in weight. It also has numerous air pockets throughout, making it even lighter. Many people prefer to think of it as the solidified froth or foam that comes from the top of the magma and lava. While this isn't entirely accurate, it does give a general idea of its formation.
The bubbles in it are caused by the expansion of gasses as it is ejected from the volcano. The size of the air pockets can vary widely and can be very nonuniform. Still, a piece of this stone that is a foot in diameter can weigh as little as a pound.
Due to the lack of density, this is also a rock that has the unusual property of being able to float on water when it is dry. It is the only natural rock to do so, according to the Mineral Information Institute. As it becomes waterlogged, the numerous air pockets fill with water. This increases the density and eventually the stone sinks. However, if it is removed from the water and is allowed to dry out, it will usually again float.
Because of the lightness of pumice, when it is ejected by the force of a volcanic explosion, it can also be hurled great distances. Heavier rock normally hits the ground sooner. For this reason, pumice is often found much farther from the volcano of its origin.
Though cinder often looks superficially like pumice, it usually contains a much higher percentage of iron and other heavy elements. It can also be filled with air pockets by the expansion of gasses. However the air bubbles aren't enough to offset the greater weight of the elements and compounds it contains.
This means that the rocks are heavy. In water, they sink readily. When they are thrown out of a volcano, the weight causes them to often fall quickly. In turn, this latter causes the formation of steep-sided cinder cones. An excellent example is Wizard Island, within Crater Lake, Oregon. Cinder cones are a common sight in the Cascade Mountains of the Pacific Northwest, some of them surrounded by old lava flows.
A cinder stone a foot in diameter will often weigh 10 to 20 pounds or more. Some of the rocks can be extremely dense, and it may even be difficult to see the air pockets it contains.
Pumice and cinder may be produced by the same volcano, and it isn't uncommon for them to have a similar appearance. They are different rocks, though. Picking up a small sample of each is likely to show the difference. If there is doubt, they can both be put in water. The pumice should float while the cinder shouldn't.
Crater Lake National Park
Volcanoes National Park
Richard Brown, Park Naturalist, ret.