I think the simplest answer is that men are viewed as more threatening and aggressive. There's also an inherent rape metaphor in how Slashers attack their victims, and a fascinating role reversal when the Survivor Girl (inevitably) uses the Slashers weapon against him. See Behind the Mask: The Rise of Leslie Vernon for a nice distillation of Carol Clover's Men, Women, and Chainsaws, which is an incredible read.
Which brings me to 2004, and the rise of a horror icon that would reign at the box office every Halloween season for seven years. Lacking a new Slasher icon, rather a resurgence of remakes giving the old Slashers a nice new gloss, Jigsaw takes the title for having appropriated elements from the Slashers while adding a 21st sensory flavor.
What The Ring and Saw do have in common is a displacing of blame from the attacker to the victim. Whereas the human personalities behind Ghostface turned out to be frauds with a false philosophy, there are very clear and consistent rules that must be followed with both Samara and Jigsaw. In the former's case, if you watch the tape you have seven days to live. If you make a copy of the tape and get someone to watch it you're saved, but you've jeopardized someone else's life. With Jigsaw, he never actually physically kills anyone for the entire series. Instead, he passes judgment on people and places them into circumstances in which they can potentially kill themselves, or in some cases kill someone else. This is all hidden behind a holier-than-thou self-help philosophy in which Jigsaw is actually helping his victims become better people.
This is, as mentioned above, totally aping John Doe from Se7en (1995). What Andrew Kevin Walker and David Fincher knew, however, is that no one has the right to judge others like John Doe and Jigsaw do. That's why John Doe is kept anonymous, and ultimately dies at the end because he freely admits he's no better than his victims. Leigh Whannell, the writer of the first movie, however, does not have the grace or elegance of those two gentleman. What is created in the form of Jigsaw John Kramer, is a man that thinly disguises vengeance with righteousness. Nearly all of the people put into traps over the course of seven movies have a personal connection to Jigsaw, they've wronged him in some way. A former engineer and toymaker that nearly dies in a car accident and then is diagnosed with cancer, he positions himself as entitled to payback. From imdb.com, 'As stated by the producers of the third film, and Director Darren Lynn Bousman, Jigsaw is not considered an actual serial killer, but a "scientist" who is determined to initiate the survival instinct in his "subjects", believing that humanity no longer used its instinct of survival.'
He is a malevolent product of our times. He's also an even more blatant rip off of the little-remembered Phone Booth (2002), in which Keifer Sutherland traps Collin Farrell in a phone booth with the threat of being shot all because Farrell is a cheater. In the end Sutherland gets away, but commends Farrell for fighting for his life.
Aesthetic - Jigsaw is encompassed by three personas, although it's unclear how each informs the other. John himself wears a red hood and cloak, giving him the appearance of a monk or Darth. The doll, Billy, has a ghoulish appearance with white skin, black disheveled hair, a business suit, and red spirals on his cheeks that hint at the labyrinth the characters have been trapped in. He also rides a tricycle, and speaks in a baritone version of John's voice. There's also the pig mask that appears to only be worn by Jigsaw's apprentices, but I'm unclear of the logistics. John himself may wear the mask at some point, but there's so much retconning later in the series it's hard to keep track. The pig mask implies that all humans (at least those not John) are animals or meat for the slab. It's fascinating that although Jigsaw uses technology, it's all very outdated like VHS and cassette tapes. There's an implication of DIY here, although it's never explained how an out of work toymaker could get the money together to build these over-elaborate traps.
Weapon - Although Jigsaw does slash a cop's neck in the first movie that cop does not die. Jigsaw does not actively kill anyone in all seven movies, but his traps do. The traps are a symbolic representation of what Jigsaw sees as a flaw in the victim's life. We know this because Billy spells it out pretty clear in a shtick that is laid out in the first movie and is stuck to relatively consistently. The traps themselves are generally set in dank, grimy basements, and as they are torture devices there's a hint of the medieval rack or possibly the Inquisition in these settings. Most are in dark places underground, hinting at the underworld or hell. They all are washed out, lit by shocking fluorescent lights, and hint at death and decay.
Modus operandi - As has been spelled out, Jigsaw (or an apprentice) kidnaps a victim and puts them in the trap, with the hope that the victim overcomes the trap and has an epiphany with the experience. Several survivors (including Amanda, Hoffman, and Dr. Gordon) buy into Jigsaw's philosophy over the movies, and even those that don't become full-on apprentices are shown to be enlightened. Only one counterpoint is offered, a woman forced to cut her own arm off in the sixth movie that rejects any lesson Jigsaw hoped to teach her. At the beginning of each trap, Billy tells the victim he wants to "play a little game" over a tv screen or tape, and that it's up to the victim whether they "live or die". In the earlier movies John also nabs pieces of flesh from the victims, which is why the media labels himself the Jigsaw killer. This is mysteriously abandoned in the later movies because the writers are lazy and don't give a fuck.
Jigsaw has taken Ghostface's phone calls, even referring to the life-or-death situation as a game, and upped the ante. The video screens may even be borrowed from the tape in The Ring, but it's not blatant enough to be sure. Whereas Michael Myers reveled in stalking his victims and Ghostface liked to make them squirm before attacking, Jigsaw uses technology to monitor his victims' every moves. This could be an attempt at meta-commentary, with Jigsaw as the director and the "game" being a scripted movie that Jigsaw has orchestrated, but maybe I've thought about it too much.
What bothers me the most about this movie series is the filmmakers' claims that Jigsaw is in the right. It's having your cake and eating it too, because the masses want guts and blood but they don't want to feel guilty about their urges. When a character dies in Saw, the rabble justify their enjoyment with "well they didn't fight hard enough" or "they didn't want to live." These movies cheat, because very rarely do any characters actually survive, and even in the later movies (when Amanda and then Hoffman take over) the traps become inescapable. However, by that point the formula has been clearly established so the audience doesn't even notice the trap is inescapable because they've become so accustomed to victims never escaping.