Difference between Organic Bread and Fresh Bread

Fresh just implies that the bread was never frozen (or canned, irradiated, salted, pickled or otherwise preserved, but those almost never apply to bread).

Organic, at least in the US implies following a set of FDA guidelines regarding prohibited methods or techniques or ingredients in producing the product.

From the USDA's Labeling Organic Products Fact Sheet:

Organic products have strict production and labeling requirements. Unless noted below, organic products must meet the following requirements:

• Produced without excluded methods (e.g., genetic engineering), ionizing radiation, or sewage sludge.
• Produced per the National List of Allowed and Prohibited Substances (National List).
• Overseen by a USDA National Organic Program authorized certifying agent, following all USDA organic regulations.

These terms are not mutually exclusive. A given loaf of bread could be both organic and fresh, organic and not fresh, fresh but not organic, or neither fresh nor organic. The term "fresh" also has both technical definitions used by government agencies and commercial producers as well as a variety of non-technical definitions used commonly.

Technical Definitions

Fresh: Suggests or implies that the food is unprocessed, means that the food is in its raw state and has not been frozen or subjected to any form of thermal processing or any other form of preservation (US FDA Fresh Definition). From the FDA definition, there can be no such item as "fresh" bread, but it might be reasonable to claim that an item is "freshly baked".

Organic: Implies that a particular food has been produced with specific restrictions on methods, techniques, or ingredients. The particular restrictions vary depending on the certifying agency or country of origin. Common restriction might include the use of pesticides, genetically modified organisms, fertilizers, or antibiotics. (USDA Organic Program)

Common Use

In common use, a fresh loaf of bread is one that has been baked recently (usually within a day). "Fresh" may also imply that the bread doesn't contain preservatives, emulsifiers, stabilizers, or other additives because many of these ingredients are not necessary for bread that is meant to be eaten within a few days. Though this may be an implication of websites or individuals extolling the merits or virtues of eating "fresh" bread, there is no guarantee that bread purchased as "Fresh" or "Fresh baked" will not include these ingredients.

Effects on Bread Quality

Most people would agree that, with a few exceptions, fresh bread has better flavor and texture than non-fresh bread. Some sourdough breads do not reach their flavor peak until 24-48 hours after baking.

Whether breads made with organic or non-organic ingredients are better in terms of flavor and texture is much less certain for a number of reasons. First, an organic designation does not guarantee that ingredients are high quality. There are high quality ingredients that do not meet the organic standards. Second, baking technique has an incredible amount of influence on both flavor and texture.

Fresh bread should have been baked that morning, or in the wee hours of the morning (middle of the night), but certainly not more than 24 hours ago. "Fresh" in terms of bread, is no indication of its contents. It may or may not be organic. It may or may not contain preservatives. Fresh in the context of bread indicates only its age, not its contents. (I think some of the other answers are confusing that word with the way it's used in connection with meats and vegetables.)

Organic bread means that it doesn't contain artificial ingredients, or ingredients that were irradiated, etc. Organic bread may contain natural preservatives. It also means that the wheat did not come from a GMO.