I would say geldings and mares are definitely different. Are men and women different? Yes.
However, each individual horse is incredibly different as well.
I was once in a barn where there were three mares and twenty-three geldings. One of the mares was mine. I couldn't believe so many people were that biased towards geldings over mares.
I'll take each of your words one at a time:
Sensitive - I believe sensitivity is much more breed related than gender related. I've met some mares that wouldn't notice if you hit them on the head with a 2x4. Figuratively speaking obviously.
Difficult to train - I find absolutely no difference in how a gelding learns vs. a mare. I don't believe this is gender related but again, some individual horses might not be as bright as others.
Stubborn - Not related to gender. Then again, I don't find stubbornness to be a common trait in horses. Instead I find it to be a common trait in horse trainers who keep on persisting when a horse doesn't understand what they want.
Inconsistent and moody - Some mares come into raging heat and this can affect their performance. However, a well-trained mare can usually still perform well when in heat. I say usually since one of my mares has one or two days a year where she is unable to stop urinating every 5 or 10 minutes, and those are not good days to ride her. Honestly though, it is hard to tell when most mares are in heat or not.
I personally bond better with mares. I like female horses and male dogs.