That said, science has shown that neanderthals' brains at the time were less developed in the areas associated with social interaction, etc, which means they'd be starting from a somewhat different place (although, if they continued to exist, they would have evolved over time)
The trick to that is that, for most of the timescale where the two species overlapped, homo sapiens was apparently ALSO less developed than modern expectations. The two species (which may actually only be two separate branches of the same species, according to some modern interpretations) coexisted for tens of thousands of years in relative equilibrium, which would have been more or less impossible if sapiens were significantly more advanced mentally than Neanderthals.
Generally speaking, what we think of as "modern" humans in terms of brain capacity and thought processes didn't exist prior to around 70000 years ago. At which point we sort of kick-started into overdrive, moved into Neanderthal regions and wiped them out and/or subsumed them, spread across vast areas that would previously have been inhospitable (ie, the upward drive into Siberia and ultimately over into North America), and started developing the techniques that allowed us to reach islands (and to island-hop our way to places like Australia).
If we're imagining a scenario in which Neanderthals could somehow survive and compete, it's not impossible to assume they could have undergone a similar shift, even if they didn't in the world as we know it. It's also possible they could have undergone said shift while sapiens didn't, and we wouldn't be here now.
I'm talking apples-to-apples. The Smithsonian article compares prehistoric man's brain to the neanderthal's and found that the development of certain parts was very different, which may be one reason why sapiens were able to succeed. ---