The active ingredient is the chemical that is reported to provide the biological effect and therefore regulated by various governmental agencies.
The adjuvants (formulation components, inert ingredients) are the chemicals added to enhance the effect of the active ingredient but are not regulated.
The inert ingredients may be the critical components that make the active ingredient a biological effect.
Monsanto has a product called Roundup that consists of the active ingredient (glyphosate) and inert ingredients. In advertisements, Monsanto has shown a person spraying weeds, and the weeds dying in a day or two. Glyphosate does not work in this way. Not ever. What the formulation does is to burn the plants with a high rate of surfactant to give high visual impact. Many plants would recover following defoliation by surfactant burn. The glyphosate prevents this from happening. The formulation components are critical for product performance.
The inert ingredients can be grouped into several categories: Spreader/stickers to help droplets adhere to the plant surface and more uniformly cover the surface. Spray modifiers to make a more ideal droplet size during spray application. Protectants like UV blockers to reduce photolysis. Humectants to reduce evaporation rates of a water diluent. Emulsifiers to improve mixing of oil and water parts of the formulation. Stenching agents and dyes to make an otherwise tasteless-colorless-odorless material more easily detected and avoided. There are many others, some of which depend on the formulation or method of application.
Active ingredient (a.i.) is the active component of a pesticide, and is that moiety which is patented, synthesized and registered by the manufacturer.
Adjuvant is broadly defined as any substance added to the spray suspension, separate from the pesticide formulation, that will improve the performance of the pesticide. This can include everything from wetter-spreaders to feeding stimulants. Pesticidal adjuvants are wetter-spreaders, stickers, emulsifiable oils activators, compatibility agents, drift retardant, spreaders, foam retardants, dispersants, activators, surfactant, buffers.
An adjuvant is added by the user. An adjuvant added by the manufacturer is a formulation component. The formulation is also patented. Hence you get "inert ingredients" that are a proprietary blend the components of which are never listed individually.
You should add water conditioners to the list. Glyphosate is inhibited by the presence of divalent cations and ammonium sulfate is added to remove the divalent cations from the water. The nitrogen might also stimulate plant growth thereby increasing uptake and translocation of glyphosate and improving efficacy.
Stenching agents and deflocculating agents could go on the list. Probably still missing some more.
Active ingredient is pure form and toxicant synthesized. whereas, active adjuvant is used in formulation of pesticides or added to diluted solution in Sprayer tank, so this is easy to mixed and used of that and sometimes reduced Consumable active ingredient value for get of appropriate control.
Active ingredient the essential substance that produce the desired effect, while active adjuvant the material that administered with active ingredient to facilitate the activity of essential material.
@Timothy A Ebert: you explained that only the active ingredient would be regulated. This may be a bit US-centric. In fact, in many countries worldwide not only the active substances but also the formulated products (including the active ingredients and all other compositants) are tested and regulated according to the respective pesticide laws. At least in Europe, the other compositants are also regulated individually under the chemical law (REACH).