While both are excellent products suitable for almost any setting, each material has its positives and negatives.
Clay pavers are made from natural clay dug from the ground. Similar to a clay pot or mug, a clay paver is shaped and then fired in a kiln to make it solid and strong. Clay can be left its natural color (which varies depending on where the clay is sourced) or the color can be augmented by mixing pigments into the clay before shaping and firing. Clay pavers are extremely tough and durable, with a naturally slip-resistant surface. They are not prone to cracking or disintigration, often don't require a sealer (installations around a salt-water pool being one exception), and its natural color will not fade.
On the negative side, clay pavers are more expensive than concrete, and because clay pavers are brittle and therefore difficult to cut, they can also be more costly to install. Another potential issue is that pavers made from clay may vary slightly in size, due the nature of the material. However, the variance is very small, and this isn't a problem for a skilled paving installer.
While clay is a natural material, concrete is man-made. Concrete pavers are made from crushed stone, sand, and cement, usually with pigments added to create the desired color. The concrete is mixed, then poured/pressed into molds to set and dry. Concrete pavers are strong, durable, and low-maintenance, with the added benefit that they cost considerably less than clay pavers. Concrete pavers are readily available in a wide range of colors and shapes, and some are even made to mimic expensive natural stone. They are very uniform in shape/size and more easily cut than clay pavers, making installation less costly.
Unfortunately, over time, UV rays from the sun will bleach and fade unsealed concrete pavers. This may not be a problem, if you don't mind a mellowing effect. However, you can avoid fading by coating concrete pavers with a UV-blocking sealer, which will have to be reapplied periodically to protect the original color. Sealing also prevents the porous concrete pavers from absorbing any oils, grime, and dirt that it comes into contact with.