I can absolutely feel the difference with a silent or near silent shutter system compared to my dSLR. My 5D mirror slap is so obvious everyone knows when I've taken an image, and if my 1D fires of 10 FPS in a burst, it's quite obvious. But when I'm using my Fuji mirrorless, I can be within 2 feet and you can't hear it. I've been a lot of situations and still experience this where the shutter sound interference with the image capture process because the subject becomes painfully aware that I'm there and doing something, be it a bird at close range, or a bug at close range, or a person at close range.
Just a few weeks ago I was aggervated with a particular moth on a flower I was macro imaging because my shutter was disturbing it at 8~10 inches away and would disrupt what it was doing and cause it to stress and fly a bit and I'd have to move and follow it again. When I use my mirrorless, I can take images and literally can't be heard and nothing is disrupted from the noise. The difference is such an experience, quite eye opening. Same thing with birds, where I've lost lots of chances with a few spooky King Fishers that heard my shutter one time and took off never to come back. Annoying.
Mean while, even more obvious for me, is when I'm shooting my 5D with a 35mm or my Fuji at 35mm and how my 5D is so loud and I'm very close to the subject(s) and it's obvious I'm taking photos and they can hear my mirror slapping around so it takes away that candid or natural look that I'm capturing (especially when I'm doing family photojournalism or shooting in public at close distances, which I really prefer to be close). When I added my mirrorless with the 35mm FOV, it's now dead silent, and I can be in their face with it and they can't tell I've taken a few images. And I really like that, because it doesn't disrupt the moment and I get the candid and sincere expressions I'm looking for, rather than breaking the moment and losing future moments because now everyone is very aware that there is a "camera taking pictures."
I would love a 1D with a silent shutter (a real silent shutter, not just a quieter mirror slap at the cost of FPS).