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Is there an advantage to having one over the other? Does one have a better feel, sound, or response than the other? What are your thoughts?

In my experience, gold plating feels smoother than silver, and the "clean" feeling lasts longer. The mouthpiece itself had no different playing characteristics than the same silver version.
Otherwise, the main reason for gold plating mouthpieces is to avoid silver allergies that some people have

DCottrell wrote:In my experience, gold plating feels smoother than silver, and the "clean" feeling lasts longer. The mouthpiece itself had no different playing characteristics than the same silver version.

I wouldn't rule out the possibility that the smoother feel is related to the playing difference some people notice. It wouldn't work for everyone, only if you're up against the rim in a way where surface friction affects lip movement. I guess that could be about either vibration or shifts. Just guessing.

Silver plating wears off in a few years. Gold plating wears off much more quickly. Stainless steel does not wear out, and neither does plastic.

One big difference between silver or gold plating is the grip for the embouchure. Gold plated rims feel a bit slippery when you play it for the first time because of the much smoother surface. So you need a bit more training for playing really loud, but that is not a bad thing... On the other side you can move your embouchure freely when playing fast through all registers.
For the sound I could not find a difference between unplated brass, silver plated brass or gold plated brass. It is just a matter of taste. Plastic mouthpieces offer the same grip as silver plated (at least Kelly and Brand) while stainless steel feels more like gold plated. With stainless steel I found the sound a bit brighter compared to brass or plastic mouthpieces, but I only tested it for a short time.

There are some titanium nitride plated mouthpieces out there. I have never played one. I would think they would be slippery like gold but much more scratch resistant, it's a very hard coating. Plumbing fixtures survive being scrubbed with abrasives.

""Sound production" is really not something that can be judged (between stainless steel and plated brass) unless the person doing the judging is absolutely 100% certain that a particular stainless steel mouthpiece and a particular plated brass mouthpiece feature absolutely identical interior dimensions and rim contours."

I had the nice opportunity to test my long played plastic Kellyberg together with a silver plated Conn Helleberg and the stainless steel Kellyberg. When played with the back towards the audience they were not able to hear a difference between the Conn and the plastic but the stainless steel was easy to distinquish.

Indeed it might be interesting to make some careful measurements (to a higher standard than I could manage myself.) As it happens, I repeated part of that experiment myself yesterday, as I started out playing something on my Kellyberg and switched to my Faxx Hb, and I'm confident anyone could have heard the difference. But ... I could go into it at some length, but suffice to say, not all of it was the material, likely none of it was. Though they're very similar dimensions. I'd particularly measure (if I could) the exact shape of the throat, i.e., transition from the bowl of the mouthpiece to the backbore. My suspicion is that this critical area is characteristically rounded in plastic mouthpieces as a natural consequence of their manufacture.