Spiders are not considered bugs in the informal sense. They are arachnids, which include a different classification of arthropods that have a very different anatomy than that of insects or "bugs."
The main difference between arachnids and insects is that arachnids have bodies divided into two regions, while insects have three distinct body regions. The parts of an arachnid are called the cephalothorax and the abdomen, while an insect possesses a head, thorax and abdomen. Another way the two differ is that an arachnid has four pairs of legs attached to the cephalothorax, whereas an insect has only three pairs of legs attached to the thorax. Also, no arachnids have wings or antennae.