In my not-vast-but-not-tiny experience of talking with other gardeners, I have found that the word "organic," as applied to gardening, seems to be not well understood. There's a good reason for that; the definition isn't easy or brief, but I am offering here a simplified explanation.
When we refer to organic gardening or farming, we mean growing food using a specific set of principles and inputs that are as close to the natural state as possible in a way that maintains a living soil with a diverse population of micro and macro-organisms. Spraying anything for pests and diseases is the LAST option for resolving garden problems, even if an organic-approved spray is available.
There's some significant overlap between organic and conventional gardening. Plants need nutrients, soil/support, water, sunlight, air in the root zone, and good air circulation around the leaves and stems, and those are the basics that good gardeners using both systems provide.
There are huge differences, too. In conventional gardening, soil is viewed as a substrate with important physical and chemical properties that affect how nutrients and water move through the soil. Fertilizers tend to consist of salts of various essential nutrients, which are available for uptake by plants as soon as they are dissolved in water. Many chemical options are available for diseases and pest control, and correct use of inputs (fertilizers, for example) depends on some simple math and basic guidelines.
In organic gardening, soil is viewed as home to an abundant and diverse community of tiny life forms. The physical and chemical properties are important, too, but more important is that nutrients are made available when released through the action of those microbes, fungi, and other tiny lifeforms that live in the soil. This action is, essentially, the decomposition of organic matter and other soil amendments. Maintaining the health, abundance, and diversity of this community underground is essential to having a productive organic garden. There are very few spray-on options available for pest and disease control, and those that are available don't work all that well (in general). Choosing inputs -- manures, composts, and rock powders, for example -- to maintain the abundant liveliness of the soil, takes careful thought and planning.
Looking at the differences between the two systems, and the absence of absolutes -- or simple prescriptions for what to do next -- in going organic, the big question is "why would any sane person choose organic gardening?"
Well, I can think of plenty of reasons. To start:
1. living near an ecologically sensitive area (like a stream) and not wanting to mess that up
2. wanting to provide as little support as possible for "big agriculture," for one reason or another
3. being majorly into DIY (doing it yourself), because with organic, you can
4. having small children or pets, and as a result not wanting to risk storing hazardous chemicals
5. having a serious sensitivity to a wide variety of chemicals, and wanting to be free of rashes, fatigue, etc.
6. being concerned about losses in populations of bees and other pollinators
7. having a tendency to put food in your mouth - unthinking and without washing - while in the garden, or having a child with the same tendency
8. wanting to eat organically grown food, while at the same time having a tight food budget
9. being concerned over some of the newer, systemic pesticides used on commercial crops that can't be washed off, because they are taken up inside the cells of the plants
It is difficult to just go partway organic. Using composts and manures can be a big help in conventional gardening, improving water retention/drainage and nutrient flow/abundance, but using conventional fertilizers in an organic system is more likely to have negative effects. Some members of the below-ground community of micro and macro-organisms are very sensitive to the fertilizer salts; they will do less well if conventional fertilizers are added. If the action of those lifeforms is the major source of nutrients for your garden, their doing less well will be a problem, because your crops will also do less well.
Going organic also means that most pest and disease control is done through prevention, involving crop rotations, disease-resistant plants, avoidance strategies, cover crops, promotion of beneficial insects, and other strategies that require advance planning.
This sounds supremely complicated, but plenty of gardeners seem to be managing organic food production quite well, and we are fortunate in having a lot of information and other resources to help us along the way.