Compost is an inexpensive alternative to chemical fertilizers, and it is less likely to harm sensitive roots. Chemical fertilizers can be extremely harsh on plants. Chemical fertilizers can also leave heavy metals like lead, arsenic and cadmium that can build up over time. A big shock of chemical fertilizer can also kill the very microbes that make soil fertile, dooming you to depend on using chemical fertilizers over and over again. Many chemical fertilizers are made from petroleum and other non sustainable sources. The price of chemical fertilizers seems to get higher every year due to rising oil and shipping costs. On the other hand, compost doesn’t use any oil and doesn’t require much transportation. Using compost reduces our dependence on foreign oil and boosts soil health.

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Yes, it is better to use compost instead of chemical fertilizers because by using compost, we are reusing our garbage As, Compost is nature's way of recycling organic material back into the environment. Adding fallen leaves, cut grass and kitchen waste to a compost pile instead of sending them to a landfill returns to the earth those nutrients the plants took out.

 Also compost is cheap and less harmful to than chemical fertilizers.

Compost feeds the microfauna in the soil. Chemical fertilizers are unable to do this. The microfauna, in turn, provide nutrients to plants over a long, sustained period, reducing the opportunity for the roots to be burned by excessive nitrogen.

Compost increases the tillage of the soil, making it easier to work. Heavy clay soil is lightened by the presence of compost, which allows it to breathe easier and avoid clumping. Sandy soil is bonded together more tightly, allowing better water retention and less opportunity for the surface to blow away.

Hope it help... If you want a difference between these two...
Or see my next answer hope that help... That is just this...
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Fertilizer and compost are different items. Neither is better. Compost is a soil amendment like peat moss, except it has much more. It normally contains many micro-nutrients that common fertilizers do not have. It also has live microbes fungi and bacteria that are needed by many plants to thrive. Composts may have small amounts of N,P,K (Nitrogen, Potassium & Potash) fertilizers within it, but it depends on the source material of the compost. Soil amendments change the structure of the soil so r