While Neanderthals were genetically almost identical to us, as fellow erectus lineage cousins, the baby could probably be raised the same way.

The problems would come during socialization.

For perspective:

Neanderthals have, on average, brains that have about 10% more volume than sapiens.

Neanderthals brains have a smaller blood supply, and fewer crenelations, and lower neuronal density.

That means Neanderthal brains need more volume, to have the same processing power.

The only regions of the Neanderthal brain that were actually larger though, were dedicated to processing scent, vision and motor control… not aspects of intelligence.

The smaller regions included, for example, those processing abstract reasoning, speech and language, and social skills.

To have the same processing power, those regions would need to be larger than sapiens’, but they were smaller.

So… the baby would be handicapped in those areas of intelligence and social skills etc.

Most primates continue brain development unto about 3 years of age.

A 3 year old chimpanzee tends to have related test scores in line with a 3 year old sapiens for example.

The same way a person’s genome normally turns off the ability to digest milk after weaning age, there are mutations that left that turned on, so milk digestion continues into adulthood….

… mutations resulted in brain development continuing after 3 years of age… into adulthood.

Neanderthals and sapiens shared almost all these mutations, and, up to the age of 3, the baby would probably be expected to be on track with their peers.

After that, the small differences in alleles for intelligence etc, would progressively start adding up… and the child would start to struggle to keep up.

As the differences were small, it would not be as dramatic as the difference between 3 year old vs 4 year old sapiens and chimpanzees… at all, it would be a subtle loss of ground over years…

… as the lower processing power aspects became more and more germane to their lives.

How old when first realizing that Santa being real, was impossible, getting abstract concepts, etc.

As some sapiens are cognitively challenged in the same way from time to time, there’s a spectrum of abilities, so some sapiens and some Neanderthals are going to be more, or less, intelligent than others, there’s probably an overlap where the two share abilities, for example.

After that, a Neanderthal looks a little different.

They have shorter shins, wider hips, a more robust skeleton, different shoulder construction, huge heads with huge noses, no chins, a sloped forehead, and a skull elongated to the rear, and large brow ridges.

So, not exactly most people’s cup of tea, and a child’s nightmare if wanting to just fit in at school, etc.

So, when socialization is an issue, being odd looking, and a bit slow, and having worse social skills, is where the poor kid would be challenged.

And, as topics arose that required more abstract reasoning, being capable of that, but not as good at it, that too would be a struggle.

So, being an odd looking kid with poorer social skills, the poor kid’s life would be challenging.

That said, they might simply niche into the same sort of life any odd looking person who’s a bit slow on the uptake and doesn’t always get what people are talking about would.