The composition of smoke is not well-defined. On the one hand, some smokes could be considered aerosols of solid particles or liquid droplets only (no gaseous material worth considering). These smokes, as well as other aerosols (e.g., sneezes) can be filtered well enough by using fine weave fabrics (e.g., K95 medical masks). There does not seem to be any suggestion that wetting these masks improves their performance, perhaps because the fine weave would be blocked too much, even by a thin film of water or by swelling of the fabric material.
On the other hand, it might seem that a wet mask could trap liquid aerosol droplets (assuming they are aqueous, not oily) more easily, and this might prove advantageous if the mask is less protective or efficient than a K95 mask. Such a less effective mask probably wouldn't close down so much if it became wet.
On the other hand (if you have one), the gases in a fire combustion smoke may include NO2, SO2, HCN, H2S, Cl2 (from PVC), dioxins, methanol, and many others (Ref 1). These might well be trapped better by a wet mask because they won't be "trapped" at all by a dry one.
Smoke inhalation is the primary cause of death in victims of indoor fires. Particulate inhalation could certainly cause death, but I think the gases would cause incapacitation faster (choking, gasping, collapsing) than suspended particulates.
The graph below shows chemical compositions of volatile organic compounds in smokes from various fuels (Ref 1). The prevalence of oxygenates is obvious; these would likely be better absorbed by a wet mask.
In a reverse sense, wet masks seem to prevent transmission of large aerosol droplets outward better than dry masks (Ref 2). But for COVID protection, dry masks are preferable (Ref 3). But, wet or dry is better than nothing (Ref 4). And cotton exposed to humidity shows an improvement in filtering, whereas synthetic fabrics do not (Ref 5).
In general, specific testing, against the specific smoke (or its likely composition), of the exact mask or cloth/towel/etc., would be needed to determine what would "help you breathe better". In the short run, helping you get out of the smoke would be best; perfect elimination of contaminants with overly restricted airflow isn't the best solution. Perhaps every situation is unique.
Ref 1. https://en.wikipedia.org/wiki/Smoke
Ref 2. https://www.sciencedaily.com/releases/2021/11/211122135517.htm
Ref 3. https://www.wusa9.com/article/news/verify/does-a-wet-mask-make-a-difference-or-am-i-safe-wet-mask-efficiancy-vs-dry-masks/65-5d760229-9e3e-4649-b420-969c74f219cb
Ref 4. https://www.popsci.com/science/wet-masks-covid-protection/
Ref 5. https://www.nist.gov/news-events/news/2021/03/study-indicates-humidity-breath-makes-cotton-masks-more-effective-slowing