The way I see it is this: Synthetic fertilizers feed the crops, while organic fertilizers feed the soil. To elaborate:
Chemically, "organic" means carbon-based compounds. "Organic farming" has a different and more complicated (and USDA-regulated) meaning, but ultimately this is the difference between synthetic and organic fertilizers. Synthetic nitrogen fertilizers, for example, contain some combination of ammonium (NH4+) and nitrate (NO3-), which are chemically
inorganic. Organic nitrogen fertilizers are largely protein / amino acids, which are carbon-based molecules containing a high amount of nitrogen, and are chemically
organic. Since
carbon is able to form four chemical bonds, organic molecules can be very large and complex.
The important differences between the two types of fertilzer are that the inorganic molecules are more soluble in water and are more easily taken up by plants. While this would seem to be a good thing for the plants, examining the pathway of the two types of molecules from fertilizer to plant reveals a counterintuitive effect:
Synthetic fertilizer quickly dissolves into the water in the soil, creating a large pool of nutrients that can be easily taken up by plants. The crops therefore receive a large pulse of nutrients, stimulating fast photosynthesis and growth. This growth occurs faster than the plants can produce secondary metabolites, compounds which protect the plant from pests and improve the taste of the produce. This makes the crops vulnerable to pests (as noted by John) and less tasteful/nutritious (as noted by Paul). The crops, having their nutrient needs met with compounds that they can absorb directly, reduce their reliance on symbiotic soil microbes. This harms the crops' long-term nutrient uptake ability and disease resistance. The crops are also unable to take up all of the fertilizer dumped on them at once, and the rest is left in solution which is easily leached away, reducing the efficiency of the fertilizer and causing problems (eutrophication) for nearby aquatic ecosystems.
Organic fertilizer is less soluble and less easily taken up by plants. Plants can't generally absorb organic fertilizer directly through their
roots, because the molecules are too large/complex. Therefore, they rely on soil microbes to break down the organic molecules into inorganic forms (a process called mineralizing). To aid this process, plants excrete carbon from their roots to "feed" the soil microbes. The soil microbes incorporate the nutrients into their cells, where they are resistant to leaching. The nutrients pass from the microbes into the soil as the microbes die and then back into microbes, meaning there is a small but steady pool of plant-available nutrients in the soil, while the rest is stored away. Mycorrhizal fungi can also pass organic nutrients directly to plants, which increases secondary metabolites. Overall, the plants are healthier because of slow, steady growth and associations with soil microbes.
Note that while this picture is pretty well supported by research, reality is much more nuanced. In general though, I believe organic fertilizers are better for long-term soil health, crop production, and the surrounding environment.