If you're not an egg person, the difference is probably subtle.
There is an obvious difference in freshness between supermarket eggs and farm eggs. Egg freshness is uncontroversial; as they age, the membrane around the yolk breaks down and the whites thin. Supermarket eggs are usually around 2 weeks old.
There is an obvious difference in color between supermarket and farm eggs; farm eggs have orange yolks and supermarket eggs have yellow yolks. This does make it hard to blind an egg taste test, though. The color difference comes from the feed of the chicken.
Apart from cooking up better, farm eggs taste richer to me, and don't have "off" flavors (a small number of supermarket egg chickens get feed that includes fish meal).