Knitted underwear has been around since before 1900 and though I cannot find any information about them predating 1930, I assume woven boxer shorts are at least as old. The competition between the two is usually characterized as boxers vs. briefs but I don't know if that is actually as good a comparison as knit vs. woven.
The role of underwear was originally for warmth. These days it principally provides a relatively inexpensive barrier between ourselves and our much more expensive outerwear. The advantages of the knitted versions from my point of view are their absorbency and support. Woven on the other hand is roomier and available from one's favorite shirtmaker in any of a thousand shirtings (I am told that there are men who wear boxers that match their shirts every day but I do not believe I know any of them, nor am I going to try to find out). The boxer vs. briefs comparison blurs however when one considers that there are knitted boxers these days, and someone, somewhere has undoubtedly designed woven briefs.
The choice between the two being a religious issue, far be it for me to declare knitted or woven better than the other. I will say only that I grew up a knitted man, switched briefly to boxers when I was enamored with the idea of patterned stuff like the pair in the Apparel Arts illustration from 1933, and swung back to knitted after just a few years. There I have remained ever since.
Are you a knit or a woven man?