It would seem that Neandertals, at least the ones living around Gibraltar(who were also among the last of them) liked their food variety. At least there seems to be evidence that they liked to eat marine mammals when available. Previously, it seems, a lot of people thought they ate nothing but meat, meat, meat, from woolly mammoths or some other four-footed land creature. Here, though, they apparently even ate mussels, and they weren't raw, either.
John Hawks(You have to scroll down a bit to get to his comments, though) has a commentary on the original paper, which went into considerable detail about the Gibraltar Neandertal diet.
About all I can say to this is, every time someobody comes up with a theory as to why "modern" humans are here and Neandertals aren't, a discovery like this comes along to confound people. We really don't know why Neandertals aren't around any more. IMO, the reason probably has a lot to do with their rather small and scattered population, But I'm only a Starving Writer, not a scientist, so I know no more than anyone else.