I think they both are quite similar, but with one main difference. A dream is something that you wish were true, or something you hoped the other hand an aspiration is something that you intend to pursue.
There is slight difference between your dream your aspiration. It's upto you, what do you mean by your dream and your aspiration here. In my opinion, dream is just our imagination what we see and observe in our day to day life, or when we are asleep. And aspirations are our willings, what we want from our life, what we want to became in our life. Simply we can say that "Dream" is what we see and "Aspiration" what we see by open eyes.