Black and white conversions

OK--I converted this photo to b&w three different ways, and honestly don't see much difference.


This one I desaturated:


This one I converted to grayscale:


And this one I removed the color:


I see some difference in the one that I converted to grayscale, but not much--and none between the other two.  Is it just me? Does it matter how I do it?  Are there advantages to any of them?