Plywood for sure is far more sturdy than MDF for mobile use, bumps and scratches and the occational rain shower. For a stationary indoor loudspeaker however, it appears that bracing and damping/lamination to be more important. There are differences but they are not that huge. Here is a test conducted by the German magazine Hobby & HiFi.
The setup is a dual chamber box with a driver in bisecting wall.
A board to be tested was placed in front of the driver and both soundpressure in front of the box (compared to no board) as well vibrations of the board was measured
Vibrations to the left and reduction in sound level to the right. The referense was the 19 mm MDF at the bottom. That result is inserted in the other graphs for comparison.
Not much difference between 16 mm MDF, 19 MDF and 18mm Plywood. Fire retarding concrete chipboard 20mm is better but hard to come by.
28 mm MDF is better but not by that much, 20mm stone is way better, 19mm chipboard is slightly worse than MDF and 19mm blockwood a bit worse than chipboard.
These measurements were made in the frequency -amplitude domain, I have seen measurement in the time domain and they show most effect of bracing and lamination more than the instrinsic qualities of the board.