+1 to echoist's comments.
My two cents:
I have both an LP and and SG so I guess I can comment fairly accurately... My LP has a really nice round and fat tone, but still maintaining good clarity in the high end, with stock factory installed pups. Rhythm work is nice and phat, with good punch, solos have authority and power, but not enough cut IMHO. The SG has a very mid-dominant honky tone, due to the all-mahogany construction, and the reduced body mass. It has significantly reduced sustain, compared to the LP, a lot less phat, sounds 'thinner' to the ear. I do like using the SG for doing solos and lead fills though, it has the right balance of cut and punch. But I think its way too muddy for my taste, even after changing pups 2 times... tried the SG with 57 classics, PRS McCartys and PRS Dragon II (my current pups). Also tried the newer SG Stds with the stock 490 pups, more power, punch and sustain IMHO but still very mid-rangey and honky.