Insects are a specific kind of animal (a class of arthropods) and insect is a scientific term. Bug can be used in a general sense to mean any small land arthropod (a “creepy-crawly” thing) that includes insects, spiders, and various other kinds of animals. In this sense bug is not a scientific term.
To an entomologist, however, a bug (more properly “true bug”, as pointed out by @dpworkin) is a specific order (hemiptera) of insects, used in a narrow scientific sense. To further complicate matters, many insects known informally by common names that include the word bug are not actually true bugs at all. For example, lightning bugs (aka fireflies) are a kind of beetle—neither bug nor fly.
So, depending on context, either some insects are bugs or some bugs are insects. Such is the nature of language.