Does the box under the mattress affect sleep?

My brother is throwing out a spare bed in his room, and before he throws out the box mattress, I’m wondering if I should take it to replace the slightly older one in my bed.

I guess my sleep hasn’t been good lately, and a new mattress would definitely help, but does the box mattress make any difference? Besides being a little more worn out, I don’t see much difference between my box mattress and the other one, though I guess the wooden boards may have sagged in a bit.

Any thoughts?

EDIT: His box mattress seems to be metallic under the cloth (a metal framework), whereas mine is wooden. Also, the boards on my bed that hold the mattresses are wooden, whereas on his they are metal.

Is there a difference between sleeping on wooden boards or a metal framework? Which one is better?