The main difference between the two notions is that while when something is eliminated, it stops existing, when something, for example, a substance is neutralized, it does not disappear but its effect becomes neutral. To neutralize, in other words, means to render something ineffective, harmless. Thus, in military sense, it means to make something incapable of action, or if a country declares neutral, it is inviolable during the war. In medicine, it refers to counteract an effect of a drug or toxin. On the other hand, with Latin origin, to eliminate basically mans to remove or get rid of something which is considered undesirable. It also means to omit something which is considered not worthy to include in something. While in Mathematics, it refers to remove a quantity from an equation, in Medicine, it means that the body expels waste. In sports, it means to defeat the other team.
