According to Marcia Ponce de Leon and colleagues, “Brain development is similar in Neandertals and modern humans.” They reached this conclusion after comparing how the shape of the brain case changes across the growth period of humans and Neandertals. This finding differs from earlier studies of Neandertal brain shape growth (Gunz et al. 2010, 2012).
Although Neandertals had similar adult brain sizes as humans do today, the brains are nevertheless slightly different in shape:

Endocranial surfaces of a human (left, blue) and Neandertal (right, red), from Gunz et al. (2012). These surfaces reflect the size and shape of the brain, blood vessels, cerebrospinal fluid, and meninges.
Gunz et al. (2010, 2012) previously showed that endocranial development in humans, but not in Neandertals or chimpanzees, has a “globularization phase” shortly after birth: the endocranial surface becomes overall rounder, largely as a result of the expansion of the cerebellum:

Endocranial (e.g., brain) shape change in humans (blue), Neandertals (red) and chimpanzees (green), Fig. 7 from Gunz et al. (2012). Age groups are indicated by numbers. The human “globularization phase” is represented by the great difference in the y-axis values of groups 1-2 (infants). The Neandertals match the chimpanzee pattern of shape change; Neandertal neonates (LeM2 and M) do not plot as predicted by a human pattern of growth.
Ponce de Leon and colleagues now challenge this result with their own similar analysis, suggesting similar patterns of shape change with Neandertals experiencing this globularization phase as well (note that endocranial shapes are always different, nevertheless):

Endocranial shape change in humans (green) and Neandertals (red), from Ponce de Leon et al. (2016). Note that the human polygons and letters represent age groups, whereas the Neandertal polygons and labels are reconstructions of individual specimens.
The biggest reason for the difference between studies is in the fossil sample. Ponce de Leon et al. have a larger fossil sample, with more non-adults including Dederiyeh 1-2, young infants in the age group where human brains become more globular.

Comparison of fossil samples between the two studies.
But I don’t think this alone accounts for the different findings of the two studies. Overall shape development is depicted in PC 1: in general, older individuals have higher PC1 scores. The globularization detected by Gunz et al. (2010; 2012) is manifest in PC2; the youngest groups overlap entirely on PC1. The biggest difference I see between these studies is where Mezmaiskaya, a neonate, falls on PC2. In the top plot (Gunz et al., 2012), both Mezmaiskaya and the Le Moustier 2 newborn have similar PC2 values as older Neandertals. In the bottom plot (Ponce de Leon et al., 2012), the Mezmaiskaya neonate has lower PC2 scores than the other Neandertals. Note also the great variability in Mezmaiskaya reconstructions of Ponce de Leon et al. compared with Gunz et al.; some of the reconstructions have high PC2 values which would greatly diminish the similarity between samples. It’s also a bit odd that Engis and Roc de Marsal appear “younger” (i.e., lower PC1 score) than the Dederiyeh infants that are actually a little bit older.
Ponce de Leon et al. acknowledge the probable influence of fossil reconstruction methods, and consider other reasons for their novel findings, in the supplementary material. Nevertheless, a great follow-up to this, to settle the issue of Neandertal brain development once and for all, would be for these two research teams to join forces, combining their samples and comparing their reconstructions.
REFERENCES
Gunz P, Neubauer S, Maureille B, & Hublin JJ (2010). Brain development after birth differs between Neanderthals and modern humans. Current Biology : 20 (21) PMID: 21056830
Gunz P, Neubauer S, Golovanova L, Doronichev V, Maureille B, & Hublin JJ (2012). A uniquely modern human pattern of endocranial development. Insights from a new cranial reconstruction of the Neandertal newborn from Mezmaiskaya. Journal of Human Evolution, 62 (2), 300-13 PMID: 22221766
Ponce de León, M., Bienvenu, T., Akazawa, T., & Zollikofer, C. (2016). Brain development is similar in Neanderthals and modern humans Current Biology, 26 (14) DOI: 10.1016/j.cub.2016.06.022
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