What preserves battery/CPU usage etc the most: screensaver or "putting display to sleep"?

The purpose of a screensaver is to avoid screen burn, not to reduce battery or CPU usage. To reduce battery and CPU usage, suspend or hibernate your computer.

I think best way to know is removing battery (if using a laptop) connect it directly to the AC, and use a power measurement tool. Those devices that measure how much power you are using. Just compare and you will know. However, I would say that putting monitor to sleep is the best option, I usually do that to preserve lifetime of screen and also save battery (because showing ANYTHING on screen requires power unlike sleep mode).