I tend to squeeze a man’s hand harder, because they often squeeze harder as well (women never do, as far as I’ve encountered). I voted I am a man, and I make a gender distinction in how I shake. because of that, but now I’m not sure now if just squeezing harder is included in what the OP meant by different.
We still exist, however. 
As a tradeswoman I have a much stronger grip than a non-tradeswoman. When shaking hands with another woman I exert no more pressure than I receive. I have no need to play dominance games and do not wish to inflict pain. With men, I also generally take my cue off them, though fellow tradesmen do appreciate a firm, confident handshake. Most men are intelligent enough to NOT do the crushing dominance bullshit with a tradeswoman because, even if I do have a strong female grip, my hand strength is unlikely to be the equal of a tradesman’s. They’re the guy, I’m the gal, no need to prove who’s the better man as it’s obvious whose got tits and who has a penis here.
I have occasionally had male customers who were surprised at my grip strength. They shouldn’t be. I use tools all day.
Regardless, I never do a “limp fish” even if that’s what I’m getting. I deal with a fair number of elderly customers who are either enfeebled by age or post-stroke and simply can’t manage more than a limp fish. And the one-armed guy I shake left-handed, even though I told him his hook whatsit doesn’t freak me out (I mean, hell, I put a new leather shoulder harness on it for him, if I can handle it to repair it I can certainly handle it shaking hands! Or limbs or whatever)
I am a woman and I try to shake everyone’s hands the same. Depends on what I have to work with, though. It is impossible to give a proper handshake to Princess Fish-hands. Women are also more likely to be wearing rings, which have to be taken into consideration.
I do hate it when women go limp at the last moment before contact, and drop the thumb and the approach angle. I am just never expecting it, and without thumb interlocking, I always overshoot, and just end up with a wad of fingers between ring/pinky and heel, With my index and middle fingers heading off into space , and my thumb sitting awkwardly halfway up the wrist. What the hell am I supposed to do with that?
Say something jovial like, “hey, missed, lets re-grip!” And start over.
I’m a woman and I shake everyone’s hands the same. A prefer a firm one or two pumper and a smile with eye contact. Even if I get a limp fish, I’ll still do my best to grip well enough to get a one-shaker out of it. Dammit. Stop it with the clammy fish shake!
I don’t shake a woman’s as hard. Many women wear multiple rings, or, if older, have arthritis, and I know a firm handshake would hurt. Also, my husband worked with young men at a college, so most of the men’s hands I’ve shaken belonged to healthy young men.
Woman here and I shake hands much more firmly/authoritatively with men. Don’t want them thinking I am passive due to my small size, etc.
The same for both men and women.
The same unless the female is cute . 
It’s different, but a lot more similar than the traditional “guy holds a mug while the woman’s hand goes limp” variety. The guy handshake is firmer, doesn’t last as long, and you have to watch out for dominance contests.
I don’t shake hands, I terrorist fist jab.
I shake hands the same way per gender, but I’ll admit to shaking hands frightfully when the other person is one of those “jolly fellows”, the kind who look like they were about to give you a hug rather than a handshake - IME, that’s a behavior which goes along with “grabs the fingers rather than the palm”. I know I have small hands, but most people manage to shake them without trying to insert my little finger into the other three; those booming guys, on the other hand… (the same hand, actually, but now hurting).
In any shake I react to the other shaker. If a man or woman shake soft, I soft them back. If they grip firmly, I knuckle up. If they shake up and down like a vibrator, I’ll give them the ride of their life.
Same shake, I don’t really think about it. I rarely shake a woman’s hand though, since that would be considered rude in most circumstances, so a kiss in the cheek it is.
I am actually just plain old agressive.
I am a bad woman in a male-dominated industry who uses my feminine wiles to get my way with the big boys. It is going to get me farther than any other woman here.
Feminist, I am not.
And the rigid half-smile thing. If you go to a place where high level women are meeting each other, you will see the same thing. They both know they are being summed up and don’t try to hide it.
Just curious- in what circumstances are shaking a woman’s hand considered rude? I try to avoid the cheek kiss whenever possible, which I would consider rude in more circumstances.
When it’s obvious the woman does not want to shake your hand. If she wants to shake your hand she will offer hers first or ask “Do you shake hands?” if there are different cultures between you. Trying to force a woman to shake hands with you (or touch you any other way unless it’s a life saving necessity) only marks you as a jerk at best and possibly something worse (like a rapist trolling for victims).
To be honest I’m not sure. I don’t really shake hands much anymore, I seem to do a lot of hugging. At a push I’d say that as I have big hands and ladies tend to have smaller hands I am more aware of not crushing the hand. I’d do the same to a bloke too though.
Shaking women’s hands I find pretty hit-or-miss. Some women genuinely try to grip your hand and some just sort of leave their hand out there. The latter seems more common, so on the whole I use a weaker handshake with women.
But on at least one occasion that resulted in being rudely told I had a weak handshake and being shown how to shake hands correctly. Oh, and another time I had a woman make fun of the fact that I even shook her hand at all, as though it’s exclusively a male-only gesture.
Fortunately, this “I’m not a man unless I try to hurt you with a handshake” thing is a lot less popular now than it used to be. It was a bit faddish in the 90s.
I actually called a guy out on it once.