"motsu" is more like "to hold" / "to carry", although it can be used in the sense of "to possess". "Aru" is more like "to exist" / "to be". It is used to express things that you have / possess, but it doesn't literally mean the same as the English "to have". It expresses the idea of having / possessing something by way of stating that the thing in question exists. (That's my understanding anyway ... Maybe some native Japanese speakers can correct me if I'm wrong about any of this???)

Maybe you can understand the differences much easier with the examples. 二階の部屋にはテーブルが四つある。　There are four tables in the room on the second floor. 冬休みのあと、学校にもどったとき、大きな荷物を持って来た　　After the winter vacation, when I came back to the school,I carried a big baggage with me. 持つ　is a verb,　あるis an auxiliary word.