In simplest terms, abiogenesis and biogenesis are two philosophies of origin or generation of life on Earth. According to abiogenesis or spontaneous generation hypothesis, living forms originate from non-living forms. Whereas the current accepted and scientifically proven idea is of generation of living forms from other living forms.
The idea of abiogenesis was very popular for a very long time in human history. It originated around the time of Aristotle and was widely accepted for almost two millennium. An example of abiogenesis is the rain of fishes and frogs. The idea was that a dry pond does not have fishes and frogs, after rains, we get fishes and frogs, so it must rain fishes and frogs. There are other similar examples (maggots from meat, mice from grain, beetles from dung, etc.) of abiogenesis. The common theme was lack of any observation, which is the basis of science.
It was observed by some researchers, that life originates from life instead of non-living matter. For example, flies originate from flies and not meat. Fishes and frogs originate from fishes and frog and not from rain (rain provides the moisture and water in which they live and not the life forms themselves).
The key differences between abiogenesis and biogenesis is the use of evidence, which the former doesn't have (and hence fails in experiments), while the latter is proved by scientific observations (And repeatable experiments). Biogenesis is currently accepted form of life generation.
Abiogenesis and biogenesis are both theories of all living organisms way before humanity.
Abiogenesis is the belief that living organisms came from non-living organisms, usually through some big accident (like a meteor crashing into Earth thereby causing a chemical reaction or something) but is unable to be determined and actually confirmed through experimentation.
Biogenesis, in contrast, is the belief that living organisms originated from preexisting living organisms (like cells) that was later mutated to become a significant form of living organism. This theory is more popular among scientists, and are often proven through experimentation.