Organic Fertilizers are derived from organic sources. In other words, dead and decayed plants, animals and animal parts. Non-Organic fertilizers are derived from other sources such as mined minerals or are processed and produced in a factory. One is not necessarily better than the other, they are just different.
Non-organic fertilizers are often stronger and more potent. They pack a larger nutritional punch for your plants. They are very effective and can be overused if one is not careful during application. These fertilizers feed plants directly and are often fast acting. Think of it as giving your plants a shot of espresso in the morning!
Organic fertilizers are more subtle in effect. They are full of beneficial things that gradually improve the soil as well as the overall environment in the area. Being organic, they will not burn your plants and can be applied as you see fit. They are often slow acting and feed your plants indirectly by improving the soil first. Think of organic fertilizers as offering your plants a good healthy breakfast that sticks with you all day rather than a shot of caffiene!