Why do some people have allergic reactions to insects and others do not?

For the same reason that some people are allergic to peanuts or milk and others aren't. It's partly environmental (based on your exposure to the offending item) and partly genetic. My brother used to catch and crush (yeah he was troubled) bees, and they often stung him. One day he had a severe reaction and has to avoid bees. Maybe it's karma but his doctor says he developed the allergy thru overexposure to bee venom. Other people can have a fatal reaction to their first bee sting.

Overexposure to an allergen will result in excessive production of IgE. B Cells in our body usually produce IgG displayed on their surface and very little of this IgG is changed to IgE (a process called Class Switching) in non-respondng individuals. For those who become allergic due to to overexposure the class switching (from IgG to IgE) increases, IgE is the antibody which can bind with mast cells and Basophils which contain active substances leading to allergic reactions. Depending upon the location these active chemicals (Histamine etc) can produce vasodilation, increased permeability of blood vessels, bronchospasm, excessive production of mucous which results in symptoms we see.