The larger an animal, the larger the vegetative parts of its brain needs to be. Controlling for body size (and possibly even neglecting body size * ), Neanderthals did not have larger brains than do modern humans. An even greater advantage for modern humans would be seen in terms of neocortex size: http://en.wikipedia.org/wiki/Neocortex#Evolution

The neocortex is the newest part of the cerebral cortex to evolve (hence the name "neo"); the other parts of the cerebral cortex are the paleocortex and archicortex, collectively known as the allocortex. The cellular organization of the allocortex is different from the six-layer structure mentioned above. In humans, 90% of the cerebral cortex is neocortex. [...]

The neocortex is part of the cerebral cortex (along with the archicortex and paleocortex, which are cortical parts of the limbic system). It is involved in higher functions such as sensory perception, generation of motor commands, spatial reasoning, conscious thought and, in humans, language.

* http://en.wikipedia.org/wiki/Neanderthal#cite_note-5

a 1993 analysis of 118 hominid crania concluded that the cranial capacity of H.s. neandertal averaged 1412cc while that of fossil modern H.s. sapiens averaged 1487cc.