Almost all the activities can be done on either the Zambian or Zimbabwean side. However I would suggest that any of the game activities be done on the Zimbabwe side as the facilities and game is better there.
Zambian activities are all slightly more expensive than Zimbabwean activities as is accommodation in Zambia.
An advantage for Zambia at the moment is that flights from Joburg into Zambia are cheaper than flying into Zimbabwe. However when you weigh up the packages available into Zimbabwe with accommodation and flights included there is not too much difference.
If you are coming to see the magnificent Victoria Falls, which I certainly hope is your main reason, then you just have to see them from Zimbabwe.
Zambia only has a fraction of the Falls on its side of the border and you will be very disappointed, if you don't view them from the Zimbabwe banks.
Bungee jumping and the Shearwater high wire activities are done from the Vic Falls Bridge, which is the border between the 2 countries. So access is easy from either side.
Wild horizons operations are all in Zimbabwe including their elephant and Zip wire activities.
You might think I am sounding biased as I am a Zimbabwean, but I promise it is my honest recommendation to stay in Zimbabwe, for the various reasons mentioned above.