Other chemcials we use come from plants, animals and bacteria in our natural environment. Synthetic pesticides and herbicides can be much more persistent - they don't breakdown as quickly. They can also attach themselves to the humus in the soil and remain inactive for hundreds of years until the humus breaks down. Synthetic pesticides also harm more than their target insect. They can kill many different types of insects at one time.
Organic gardeners work to control just the pest insects.
Insecticidal soap kills when it covers the insect. Horticultural oil kills by smothering the insect. Insects who come along later are not likely to be killed. But after spraying a synthetic pesticide, any insect that lands on the plant days later can still be killed. Synthetic herbicides have been found to be harmful weeks after application.