I can't taste the difference in my eggs vs a store bought, white, washed egg.
I can easily see it, though, with a more orange yolk in the fresh eggs, which most people attribute to eating more bugs.
The one thing I'd love to see however is whether we'd find the same proportions of nutrients in a more scientific breakdown too. Humans aren't necessarily able to know purely from the experience itself if something is healthier than something else for them, we're more wired to detect if something is problematic or not and also prone to a whole lot of suggestion. While the tasting test seems to show that the immediate experience isn't conclusively discernible, it doesn't tell us anything about long-term health effects (understandably so, they're not egg-researchers dedicating decade-long funding to figure that one out!).
I grew up with unwashed eggs and never gave it much thought, but now I guess that's one more thing to keep in mind (although I live in North America these days)
Somehow a little (probably incorrect) part of me likes to think that the unwashed eggs are more "authentic" and lead to better food - it's not rational thinking at all, and I'm not waiting for salmonella to hit me before I work at changing my mind about that :)
[0] https://cooking.stackexchange.com/a/67006