I don't know about any difference in taste, but bled eyes are certainly a lot less messy at the cleaning station.
In the heat of summer, I fill my live well with ice and no water. I let the fish die as soon as they're caught. In this case, I can easily cut them as soon as I catch them, and let them bleed out.
But this time of year when the water is still pretty cool, I like the fish to remain alive in the live well as long as possible - to keep them as fresh as possible, and so I don't have to stop to buy ice before I launch. But one guy at the cleaning station suggested I can leave them alive until I'm done fishing. Then cut them in the live well to bleed out on the ride back to the ramp, cycling the water in their a few times as I drive. Haven't tried that yet but it sounds like a great idea.