Many think they are one and the same and perhaps there are many who use them in conversation meaning them to be one and the same. There is a vast difference between the two terms so vast you can say they are polar opposites of one another. A horse is an animal with a very limited, small brain and it cannot absorb a barrage of commands thrown at him.
Breaking a horse is doing just that: forcing a horse to do whatever you will have him do with whatever means you have at hand that you wish to use to make that happen ..whether it be flogging with a whip or reins kicking with the stirrups yanking on the lunge line. It all results in breaking the spirit of the horse which then becomes and animal that will never be of good use to you or anyone else ever again.
Training a horse calls on using practices completely opposite of the "breaking" technique.
A horse needs to be worked with using unflappable patience. It needs to be ground trained before anyone ever attempts to get on his back. Ground training begins with helping the animal know where he is in relation to you at all times. Then it requires building a trust in it toward you for a horse by nature is a jittery animal and untrained will as likely as not bolt upon any sudden noise or even slight disturbance.
It may take weeks to train every step but once that training sticks, it is there. So take your time and do it well. It is quite like riding a bike: once you learn, you don't forget. A horse is like that. Once he learns, he won't forget, and that goes along with any maltreatment set on him. That too is never forgotten.
A horse can only learn in sequence. Teach him one thing. Then once learned, teach him what follows. It takes on a pattern for the horse. Master comes, halter on; out of the stall, bridle on, saddle on. Step by step, depending on what you are going to ask him to do. Every action comes to be learned in a series of steps leading to the culmination of whatever the action is to be. The more patience and time you give your horse to learn these steps: the greater a jewel you will ultimately have in your hands.
Without these two elements- patience and time- you will only wind up with a broken horse, and you will also have broken yourself for you will never have given yourself the opportunity to learn what it really is to build that true bond that is formed between man and horse.
Horse training vs horse breaking- nowhere near remotely the same.