sounds like only obese/depressed/ill Neandertals agreed to interbreed with Cro-Magnons. May be being that picky is what did them?
Liger largeness (and the word "liger") is specific to the offspring of a male lion and a female tiger; the offspring of a male tiger and a female lion (a "tiglon" or "tigon") doesn't grow so large. What's going on is that lions rely on their mates being other lions -- male lions imprint their sperm in a way that promotes growth of the cub (which is a drain on the lioness's milk production) and female lions imprint their eggs so as to inhibit growth. It's not an interaction between lion genes and tiger genes that produces excessive growth, it is the male-imprinted lion genes failing to meet the female-imprinted genes they've evolved to rely on.
The same phenomenon (in terms of genetic arms race between the sexes -- it's phenotypically different) occurs in humans under the names Prader-Willi syndrome (maternal genes fail to meet paternal resistance) and Angelman Syndrome (other way around).
We know from sequencing studies that they had a higher level of Neanderthal alleles than modern Europeans. Their larger brain size suggests a liger-like overshoot in brain growth given the constraints of the birth canal. I suspect Cro-Magnon women had an even more terrible time in childbirth than modern women.
Anyway all these traits are only slightly correlated with those conditions. Just because you have the depression genes, doesn't mean you will even likely have depression. It looks like only 2% get it.
They could also be reactions to modern environments. E.g. people get less sunlight, which means less vitamin D, which is linked to depression. Similar things with obesity, that would have been less common in prehistoric times, and is linked to many other conditions.
I have to say I don't like the weasel term anatomically modern humans to describe non-neanderthal people.