What is the difference between an arachnid and an insect?

That they don't fly and that they're expected to have four sets of jointed legs as adults are the two most obvious ways in which arachnids differ from insects and bugs. In contrast, insects and bugs [Insecta class] have a set of antennae and three sets of jointed legs. Spiders and fellow members of the Arachnida class don't have wings or antennae.

Instead of antennae, arachnids have a set of antenna like 'chelicerae' for feeding and defense, and a pair of antenna like 'pedipalps' for feeding, moving around, and reproducing. They also tend to have eight legs. But that's where some variability enters into the class. For example, it's possible to find four or six legged arachnids among the adult mite population.