The taste of supermarket eggs and the eggs of my parents' hens is easily distinguished if you eat the eggs the same day or within a couple of days that they were laid. Older eggs have a stronger flavour. The structure is also different. The egg whites of boiled fresh eggs are more firm and crumbly while the egg whites of boiled old eggs are more jelly. When uncooked, old eggs are more liquid and fall apart, while fresh eggs hold together (if you fry a fresh egg in a pan it is generally smaller but thicker, and if you poach a fresh egg it stays together egg instead of becoming a spaghetti). The white of cooked fresh eggs is a little more opaque, and the yellow is generally a little lighter than of supermarket eggs. Fresh eggs are much harder to peel than old eggs because the white sticks to the shell much more.

As home-laid eggs age, they become pretty much identical to supermarket eggs, though there are still some differences: they are not as perfectly identical as supermarket eggs (their sizes, colors and shell thickness vary), and they have a small black dot which is the embryo (which some people don't like -- you can prevent it by only having female chickens like most egg plants).