Functionally, chemcial fertilizers feed the plants only where as organic fertilizers tend to feed the soil which in turn feeds the plants. Chemically, all fertilizers (chemical or organic) are salts.
Despite their origins, chemical fertilizers consist of chemical crystals that have been extracted either from an organic source, from waste of some other manufacturing or chemcial process, or generated expressly for the purpose of creating a fertilizer compound. Organic fertilizers remain in their organic form and come from an organic source. They often contain particles from their organic source and/or are suspended in the organic sources particles. For example, manuers are often considered an organic source of nitrogen. While they do contain nitrogen, the bulk of their volume tends to be the animal waste that comprises the manuer which is normally partially processed grains and grasses for the products available for sale. By volume, most manuers are a lower nitrogen source than a chemical fertilizer which is "all chemical".
Most chemcial fertilizers are designed to be water soluable. One applies the fertilizer to the soil, the water disolves it, and using water as its transport system it gets into the plant. Most organic fertilizers are designed to to be turned into a usable form by "soil critters" comprised of microbes, bacteria, fungi, multi-legged animals, insects, worms, and more. In general, the goodies provided by organic fertilizer require some "soil action" which is moisture and temperature dependent.
Excess chemical fertilizer is more readily "available" to the plants than organic fertilizer. If it is too concentrated and/or there is not enough accompanying water, it can effectively salt poision or chemical burn the plants. Some excess chemical fertilizers can hang around during a season, or even from season to season. But more often than not, it travels with the water both on the surface and under the surface of the soil. In turn, this can feed unintended plants and/or generate over fertilized land elsewhere and/or result in over fertilized water ways, wet lands, rivers, and even the seas at river mouths.
In general, organic fertilizers feed the soil environment and the "soil critters" within the soil. Excess organics and/or hot organics still have the potential for some/the chemical components to be carried away through surface water run off and/or capilary action within the soil.
Chemical fertilizers can be much more charted with a direct cause and effect, specific measurement, etc... They do not rely on temperature nor climatic conditions. They are often cheaper than organic fertilizers. Chemcial fertilizers are used by home owners, land scapers, and farmers. They are used to provide food to plants so that they grow bigger and/or have better flower or food production and/or healthier. They are used year 'round. They should be used when the plant is in a growth or flowering/reproduction cycle. (Most of the plant based food we eat is either their green leaves as in lettuce, their seeds as in peas or beans, or their fruit as in tomatoes.)