There are lots of different possible ingredients and proportions of ingredients in bread, and the different techniques for making the bread. But when speaking asking about American white bread (American supermarket bread) versus French artisan bread:
French bread is artisan bread is made from flour, water, yeast, and salt – period. Good artisan bread and significant time to allow flavors to develop as the yeast slowly ferments the dough. Artisan bread, like baguettes, usually have a ratio weight of water in the dough of around 70% of the weight of the flour. This, and the baking technique, results in a crusty crust and lots of big holes (Open crumb) within the artisan bread.
American style white bread often includes milk and/or sugar, usually butter, and maybe some egg. American style white bread has a soft crust and small, consistent holes (closed crumb) within the white bread. The hydration percentage is more like 60%, and the baking temp is much lower. American style white bread gets most of its flavor from the enrichment ingredients (fat, milk, etc.) and is usually made quickly compared with French bread.