It's unlikely this can be answered one way or the other with the current level of data. Reasons for this are roughly two (or three):
There are very few bare-knuckle fights nowadays, so not enough of a statistical sample. The historical record when these types of fights were common roughly indicates acute injuries (not necessarily head though) were probably more common. But there were no good brain imaging techniques in the 19th century. And even for boxing with gloves, relevant data took a long time to accumulate. Among this data are surprising facts about gene-environment interaction:
Recent studies show that boxers with the apolipoprotein E4 (apoE4) allele are susceptible to chronic neurological deficits. Male boxers who have 12 or more professional fights, as well as the ApoE4 allele are 16 times more likely to have neurological deficits than those without the allele.
So with this in mind, one needs to account for the potential genetic bias in small samples.
Comparing MMA (in which the fighters use lighter gloves than in present-day boxing) and boxing with respect to injuries, in particular head injuries, has produced mixed results insofar, depending on the methodology. One (2015) study found that
Boxers were almost twice as likely [than MMA fighters] to sustain a concussion that involved a loss of consciousness.
Whereas another (2014) study had among its findings
The combined “match-ending head trauma” rate of 15.9 per 100 athlete-exposures in MMA was higher than previous studies found for boxing (4.9) and kickboxing (1.9).
And the more often debated weight of boxing gloves is equally inconclusive in this respect:
One study often cited about the impact of glove size and weight comes from the University at Waterloo. The 2014 paper outlines a single experiment in which a 16-ounce boxing glove and a 4-ounce MMA-style glove hit a sensor every 1.8-seconds for about five hours and a total of 10,000 strikes each.
Researchers were testing the durability of the material inside the gloves, but the experiment also provided data about the overall impact of different glove weights. The lighter gloves were found to have a higher peak force, but a shorter overall impact duration, while the inverse was true for a heavier glove.
A lower peak force and a longer impact duration give the brain more time to deform, and that’s suspected to be a key contributing factor in CTE and other brain injuries.
“Chronic Traumatic Encephalopathy is a huge area of study across all contact sports right now,” says the ARP’s Lovelace. ”But, is it better to get one big punch and get knocked out as opposed to 100 little punches where each one does a little bit of damage? We don’t quite know that yet.”
ARP board member Gelber adds, “Longer fights end up having a higher number of head strikes. These sub-concussive strikes don’t necessarily cause a concussion or knockout, but they have a cumulative effect.”
The relationship between boxing gloves and brain injuries was a key point in an oft-cited paper called The Boxing Debate, which was published by the British Medical Association in 1993. The paper clearly points out the lack of evidence that gloves prevent head injuries: “The introduction of measures intended to reduce the force of blow to the head are of little practical value if the minimum force needed to sustain either chronic or acute brain damage is not known.”
And as PoloHoleSet suggested, the matter of headguards:
Headguards first were put on boxers in 1984, largely in response to the televised death of Duk-Koo Kim, a professional boxer who died following a world lightweight title fight with popular Ray “Boom Boom” Mancini that was broadcast live on network television in the U.S.
From AIBA’s formation in 1946 until the 1984 Olympics, no headguards were worn. They were then worn from 1984 until AIBA opted to remove them on June 1, 2013. [...]
Retired cardiac surgeon Dr. Charles Butler, a former president of USA Boxing and the chairman of AIBA’s medical commission, used the two groups to compare the rate of concussions.
The results were astounding. Butler examined 28,802 rounds, in which headguards were worn in slightly more than 14,000 and in which they were not worn in slightly under 14,000.
The results: a 43 percent reduction in concussions in bouts in which headguards weren’t worn versus bouts in which they were.
In other words, Butler’s study showed a boxer was far more likely to sustain a concussion if he wore a headguard than if he did not.
Not everyone was convinced though:
That move wasn’t met with universal praise. Dr. Robert Cantu, a neurosurgeon and noted expert on concussion, condemned the move in an interview with The New York Times.
Cantu told the paper AIBA removed the headguards to appeal to fans who are eager to see the trauma on a boxer’s face.
“The idea that headgear prevents concussions is ludicrous to begin with,” Cantu told The Times. “It would be great if it did, but to say that taking it off will lead to fewer concussions doesn’t make sense, either.”
The 2012 International Conference on Concussion in Sport’s report was dubious about a headguard’s ability to prevent concussion.
And more on that
In 2013, the Amateur International Boxing Association (AIBA) introduced a rule banning headgear for male-senior open class boxers during competition. The AIBA has defended the rule change as motivated by safety and supported by internal unpublished studies. As a result, in 2018, the AIBA plans to universally prohibit headgear in competition: for all competitors (male and female), all ages and all levels. [...]
Through the media, the AIBA has referenced an internal study showing reduced injury following headgear removal . Without peer review, however, these claims are problematic. A second study (peer reviewed) referenced by the AIBA  to support headgear removal found a short-term increase in bouts stopped at the referee’s discretion after the introduction of headgear. This observation, however, may have resulted from particular caution following ring deaths in pro-boxing that immediately preceded headgear use , since the observed increase was temporary. Instead, along with a reduction in knockouts, this study ultimately found that headgear and computer scoring increased safety in all aspects of the sport . In a recent analysis, the impact of punches was systematically examined both with and without AIBA-approved headgear. A significant reduction in impact was observed with headgear, and it was concluded that the use of headgear can “play an important role in reducing the risk of concussion” . Related studies have produced similar results for the reduction of both linear  and rotational impact  through the use of boxing headgear. In the Zurich Consensus on Concussion in Sports, the authors recommend sport-specific studies to develop protective headgear that is designed in response to the particularities of each sport . This is supported by a recent study that highlights variable protective performance of different types of headgear . In fact, the lead author and senior author from the Zurich Consensus have levelled criticism directly at the AIBA for the headgear ban and for not participating in international discussions on concussion and athlete safety .