I wouldn't be surprised to find that the SGs and LPs with thicker necks stay in tune better than the '60s profile ones, just because a change in string tension (from the string eventually easing into its proper spot after you tune it) is more likely to twist the neck slightly and throw the others out of tune. If your LP has a chubby neck that might be the difference.
Personally I hate thick necks on guitars, and avoid the '50s style necks like the plague. The neck on my Explorer is fat enough for me ... once you get used to the '60s neck on an SG you don't want anything else. That said, I've heard guys rave about their baseball bat necks on their Teles, about how lucky they were to get one with that neck etc. Horses for courses