I have fished it flipping shallow vegetation and have done well. Flipping side by side with someone flipping a traditional tx rig I have picked up some bites behind them I think I might not have doing the exact same thing as them.

No I have not. I just stick with the traditional drop shot rig and that works well for me.

What does it do better than the standard drop shot rig?

In my opinion, I would say that the Tokyo rig is more capable of keeping the bait closer to the bottom than a traditional drop shot. When I'm fishing a drop shot, I will typically keep a leader length of 10-24" between the hook and the weight. I also will only use light wire or wacky style hooks on a drop shot rig.

The other advantage I have found with a Tokyo rig is pulling through cover since your dealing with a heavy style hook and your "leader" is stainless steel wire. You will also not be using an exposed hook when fishing the Tokyo rig.

Now I know that you can manipulate the different factors in each setup, but I would have to say that in short: If you have a muddy/silty bottom and fish are eating off the bottom, the Tokyo rig will present your bait just above the bottom so it doesn't sink into it.

I have been fishing it on braid under mats and in reeds. Places a normal drop shot would be too light.

I see the Tokyo rig shining best as a hybrid between a heavy drop shot with braid and a wobble head (Biffle head).
I see it as a moderate speed moving bottom bait and a drop shot as a more stationary presentation (although I do slowly drag a drop shot a lot)

I've tried it pitching/punching and didn't like all the hardware (split ring, swivels, and wire) that can get caught in the vegetation.

I will stick with the dropshot!