In 40+ years I've had mares, geldings and stallion. A good horse is a good horse and (knock on wood) to date I've never met a problem horse that couldn't be turned around with enough time, patience and the right approach for that horse. But that's not the point of the question.
The differences I've noticed with the horses I've owned or with the family/friends.
Qualifier: All things being equal......
Geldings are often the easiest to "always" deal with. Tend to have the most days with the least amount of "spunkiness" in them. Will almost always be willing to accept any rider. And if any horse is likely to become a "plug" it would be a gelding.
Mares in some cases can have their "marish" days although I'd never owned one that did until a few years ago when I got my current 5 year old. I'd been round them, but never owned one that got like that. They have bonded quicker and virtually everyone I've had has become very protective. I've had mares actually make moves to defend me from what they perceived as a threat (e.g. Put themselves between me and the "threat" even prepare to attack.) As most of you know mares lead the heard. If you have a mare that has the "lead mare" personality they can get rather bossy and it requires a little work to get them to accept that they can boss the horses, but you are their boss. If they are not the "lead mare" type then they tend to be wonderfully polite and easy once trained.
Stallions are like mares in that they will always have spunk in them, but they tend to have a bit more than mares. They are not good for novice riders in case when they are ridden near mares. I'd have let anyone ride my first stallion in his pasture, but out on the trail it would have had to be someone experienced. He was lovely and well behaved, but you couldn't take a nap and daydream while riding him. They are hard wired to mate. If you're riding him out and about and a mare in heat is nearby it's vital that the stallion be well trained and that accepts that you are in control. I loved riding my TWH stallion and he was a wonderful horse, but I'm unlikely to ever get another. They are powerful, head strong and like mares are a two edge sword, but with sharper edges
Both mares and stallions are more prone to becoming a "one person" horse (use to be a one man horse, but I realize that there are many more women riding today than there use to be
) if not taught otherwise during training and with some stallions/mares it can be a real challange to correct that (stallions more so than mares)
Today I own only mares by choice. In my years of actually owning my own horses (starting around 1971) 80% have been mares (at one time the % was smaller). Obviously I've grown to prefer mares, but it is a matter of choice, because I like connection I get with them. You can get sort of a simular connection with a Stallion, but it's a bit different. Probably accounts for why both can easily become a one person horse. My father prefers geldings and has never grasped why I've moved to having only mares. He accepts that they do become more attached (geldings can become attached to, but it often seems a little different than what you can get with mares or stallions), but he prefers less spunk and easier so he would tell you geldings are the best. Neither of us would say not to have a mare or gelding and we appreciate a good horse no matter the gender. We have our preferences for our reasons. It's really about what "you" want. Gender (or being gelded) is not what makes them a good horse, but it does bring certain characteristics that you might prefer (which is what I believe you were asking about
) I've loved every horse I ever had, not matter what they were and the saw the good and bad in each of them.