
From Ryan Whitwam at Extreme Tech:
Scientists have known for years that our Neanderthal cousins made use of fire to cook and make tar from birch bark, but we didn’t know much about where they got the fire. Were they simply at the mercy of mother nature, collecting fire from lightning strikes, or could they start their own fires whenever they needed it? Archaeologist Andrew Sorensen and his colleagues now say Neanderthals knew how to make fire, and they came to that decision after making fire themselves.
Sorensen and his team at Leiden University suspected that flint tools often found at Neanderthal dig sites held the answer to early humanoid mastery of fire. Any place Neanderthals lived, archaeologists are likely to find a type of flint tool called a Biface. These hand axes were used for everything from chopping wood to skinning animals, and the researchers believed starting fires were also part of the feature set.
At any rate, a researcher could generate a shower of sparks using the flint tool, which is usually faster than waiting for lightning to strike.
What we can say is that Neanderthals would have had the necessary tools to start their own fires, and modern research suggests these hominids were as intelligent as modern humans. More.
Hmmm. Surely, by now, Neanderthal intelligence is well established. If there was a Missing Link, the Neanderthals aren’t that link.
Also: from Fiona McMillan at Cosmos, on the same find:
Any flint tools that had been discovered appear to have been used for other things, such as animal butchering.
Nevertheless, there were hints that Neanderthals had an interest in fire making. For example, there is evidence that late Neanderthals collected manganese dioxide (MnO2), a mineral that when crushed to a powder can lower the combustion temperature of woody tinder, making it easier to light.More.

Note: Curiously, the ancient Greeks considered the starting of fire so important that the demigod Prometheus was banished from among the pagan gods and chained to a rock, for teaching the skill to mortals. Clearly, it was considered a sort of watershed in human development.
See also: Researchers: Neanderthals used fire to forge tools 170 kya
Neanderthal Man: The long-lost relative turns up again, this time with documents
and
A deep and abiding need for Neanderthals to be stupid. Why?