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.