In addition, even if there is a real effect here, it is so small, and the interactions so complex, that it's probably useless.

For example, even if this neanderthal gene happens to causally affect depression by 1%, that pathway might be something like

gene => slightly wider shoulders => that body structure is slightly more or less likely to lead to certain life outcomes, like marrying early or getting promoted or getting targeted by pickpockets => change in depression risk

As a result, the gene might currently lead to a 1% increase in depression risk, but in a few years it might do the opposite, if things change in the surrounding culture and those intermediary steps behave a little differently.

1% is a lot for a complex trait like depression. All genetics combined predicted/determines ~37%, and the part specifically attributable to SNPs is ~32% (http://www.ncbi.nlm.nih.gov/pmc/articles/PMC3404250/). So that one variant is explaining more like 3% of the genetics; another 32 such variants and about all the possible genetics is known! And they could potentially lead to major insights by looking at what the Neanderthal variant does differently from the regular Homo sapiens variant does; a recent example of this was synaptic pruning & schizophrenia.

They used a validation set of 14,000 people. Any correlations found in the first group is checked against the second group to make sure it's a real correlation.

Psychiatric disorders are commonly found to be influenced by a multitude of genes (SNPs, CNVs, etc.), often dozens or even hundreds. Each genetic factor accounts for only a small part of the difference between affected and normal groups. Frequently, perhaps mostly, the proportion is <5%, often as low as <=1%. As the article says, there are many genetic and environmental factors involved in complex conditions like depression or schizophrenia, so these low levels of effect are unsurprising.

Here's one article (and there are a great many more) describing such a finding: http://www.ncbi.nlm.nih.gov/pmc/articles/PMC3923972/

It's also discouraging that they support their finding by saying "The significant replicated association of Neandertal SNPs with mood disorders, in particular depression, is intriguing because Neandertal alleles are enriched near genes associated with long-term depression (5)".

"Long-term depression" (LTD) has absolutely nothing to do with "depression". LTD is a mechanism of synaptic plasticity, where the strength of the synapse is reduced ("depressed").

I'm a neuroscientist, and I checked ref 5.