I would think the 3/4" plywood would be stronger. The grain of the plies are supposedly crossed, and the glue is supposedly stronger than the wood.
What kind of weight are you talking about? It may depend on what you are building. Flooring can be made with 3/4" plywood but they will have reinforced underneath with 2x4. Likewise, a bookcase can be made with 3/4" plywood but the shelves only need to be reinforced in the back with a groove and the front with a solid piece of wood to cover the plys.
This is for the coffee table equivalent of my shadow box side table. The side table is smaller, but is currently holding well over a hundred pounds on the inside of it. The coffee table, being longer, will most likely get even more put into it. I am planning on building it 2 inches shallower, so this may keep the really big stuff out of it, but it will still hold a lot of weight. The plywood is the bottom of the box.
Debating these options:
1. Leave it a solid piece and secure every 4 inches.
2. Leave it a solid piece and run 1x4 supports under the bottom (will be seen from underside above bottom shelf).
3. Cut the plywood to smaller pieces and join to 1x4s maintaining the original over all size but using different types of wood and more screws. (Will be seen on top)
Which of these options would be able to make me least nervous about putting anything on the bottom shelf that I don't want squished?
I would go with option 2 or 3. With option 1, even if the plywood seems OK initially, over time the middle will sag.
I looked at your side table project and it appears you are using pine and a pine veneered plywood. Have you considered making the bottom out of 3/4" Baltic Birch plywood? It has less voids in the inner plies and is therefore stronger. To maintain the same look as your side table you could glue a piece of 1/4" pine plywood to the top of the Baltic Birch. This would give you a 1" thick base to your shadow box table.
I went with Option 2. The bottom is holding well, haven't put too much weight on the bottom shelf yet, but expect at the next family party for some little munchkin to be crawling around on it.
research torsion box design it will hold a lot more weight and weigh less