What is the difference since flexitarians and omnivores both eat the same foods?
Great question. There may be more than one answer, but I think the most simple and basic difference is one of belief and philosophy.
Omnivores eat both plants and animals. Often their diet is based on animal products including fish as the main course. They may also believe that they require these foods for protein and a strong healthy body.
Flexitarians, on the other hand, generally eat a vegetarian or plant-based diet. Reasons for their dietary choices include the common vegetarian issues, one or more of which, they endorse. (Examples are efficient use of natural resources, not wanting to contribute to pollution of the environment, animal welfare, health and wellness concerns and so on.)
So the focus of the omnivore and the flexitarian is different. The flexitarian is almost vegetarian...but not quite for whatever reason.
The omnivore regularly eats meat and/or chicken and/or fish. Those with flexitarian philosophy may on occasion, once a week or once a year, choose to eat meat, chicken, or fish. The decision may be due to social circumstances or cravings.
The term flexitarian comes from the flexibility regarding food choices. Another example is the individual who would not personally choose a plant-based diet,but lives with one who does. Outside the home, this flexitarian may be an omnivore who is considerate of the beliefs and values of a housemate or significant other.
Vegetarians and vegans do not eat animals, birds, or fish because it would be a violation of basic values they choose to honor. Food cravings or social situations aren't considered reasons to violate their values.
Flexitarians have individual and personal, but different, value systems to guide their food choices. Not better. Not worse. Different.