Just about anybody knows the difference between a home grown tomato and a store bought tomato is like night and day. The home growns have so much more flavor. The store bought tomatoes are virtually tasteless.
The same can be said of eggs, especially if the chickens are allowed to peck about on their own or if not, they are fed extra greens from the garden.
In my own experience I haven't noticed a difference in the taste per a specific chicken breed. Over the years I've had a few different breeds and I tend to rotate between brown egg layers and white egg layers. Again, this isn't due to any taste difference but merely due to my own need for something different now and then. Kind of like how I let my hair grow long and then cut it all off for a few years, let it grow out again, etc.
Because I like to have different, uncommon breeds I waited a long time before I ever got a white leghorn. I thought they would be skittish, fragile birds, but I have found the opposite to be true. They are probably the best egg layers we've ever had, consistently out-laying any other breed we've ever had. And so far they've remained healthy too. I'm not sure if that's because the weather has been milder in the last few years (no heat strokes) or if the breed is hardier than I thought.
Store bought Grade AA "large" egg on the left, home grown white leghorn egg on the right.
It's still fun to have an uncommon breed running around, and when my four remaining chickens go to the big roost in the sky they'll most likely be replaced with brown egg layers. That's just my cyclic thing. Since we can't possibly use all the eggs from a minimum order from a chick hatchery (usually 25 is the smallest order) I just get my chicks from the local feed store. I once had about nine hens at once and we were overrun with eggs. It was like growing zuchini. I couldn't make enough pound cake, or have enough boiled or fried eggs for breakfast, or make enough egg salad for sandwiches. I even felt bad about selling them to a lady at work for a dollar a dozen and ended up giving them away to whomever brought me an empty egg carton.
Home grown eggs usually have a darker yolk, sometimes even almost orange in color. (The store egg is on the left again.)
While it's probably not that economical to "grow" my own eggs, especially without a rooster so I don't have to buy replacement chicks every few years, the joy is in the knowing. I know what the hens are eating and I know they aren't getting artificial stimulants to produce more eggs. I know they aren't being abused and I try to give them a good life. In return all I ask for is a fresh egg every other day or so.