Clay, or terra-cotta, pots are heavy, which helps even substantial plants resist tipping over. Clay is porous, so oxygen remains available to roots, but moisture evaporates rapidly, so plants must be watered more frequently. As clay pots age, algae appears on the outer surface, as well as mineral salts from fertilizer and water. Plastic containers retain moisture better than clay, which is an advantage in hot or dry climates but a disadvantage if you tend to overwater plants.
Plastic pots are less expensive and readily available in many designs.
They're lighter than clay pots and may topple if plants are top-heavy. A standard-shaped pot is a good choice for most plants. Azaleas have shallow root systems and thus don't need deep containers. Azalea pots are inherently stable because of their low center of gravity and thus less likely to tip over. And because less pot is visible, the plant makes more of a visual impression.