I will tell you how they are heard in common use. Haul is much of the time used to mean carrying something as if it were cargo in a truck or trailer. It is usually completely moved without using the object in any way. It is rare that it will touch the ground at all. Pull has the meaning of dragging something on or along the ground behind. Yet this can mean on a trailer or sled which means something similar to haul. Draw usually is used to mean pull but more often meaning to pull something towards oneself and many times out of something else. To drag is to pull or draw something, but specifically along the ground. It makes contact with the ground or floor and is moved. I hope this helps Mat

Nothing in a sense. There are very subtle difference. They all mean to pull, but are more descriptive than the general term, "pull". Haul implies the use of great force, or the pulling of a large object. - I had to haul all my belongings across country. Draw is usually used with a animal. Horse-drawn carriage. Drag is to pull something that scrap the ground, that is not moving easily. I had to drag myself to work. Hope this helps!