I think walnut is pretty dense, probably a bit heavier than mahog.
I don't remember The Paul being any heavier than most regular LPs; it's a thinner body after all.
Wouldn't surprise me if they averaged a bit heftier than similarly-sized LP Specials though.
And The SG probably would've been a tad heavier than most regular SGs, being the same size (I think).
Of course that's speaking in general terms - wood can vary enough that there may have been exceptions.
Tonewise I think walnut likely did add some fullness & depth to the sound.
The Paul sounded pretty big even with T-tops which are bright and not especially fat or rich.
A long tenon likely helped its sustain & liveliness a bit too.
I don't believe tenons affect tone much, but IME long tenon necks tend to feel more alive.
I was in a band for four years with a guy who played a The Paul. Never played a The SG, at least not that I can remember.
By '79 or '80 when these models came out I already knew SGs just don't match up well with my body geometry.
Wasn't the Firebrand series walnut too? Or am I misrembering?