"Heart" and "soul" are similar, but slightly different concepts.
People's idea of the soul differ greatly, but it generally refers to the entire inner person - not our physical body, but everything that makes us who we are. Where people's ideas differ is generally what happens to this soul after death - but in life, it represents everything non-physical about a person - their thoughts, their feelings, their memories, everything "spiritual".
The figurative heart represents our feelings, and is generally separate from our mind which represents our logical thought processes. People sometimes speak about a conflict between our mind and heart - that is when the logical choice, weighing up pros and cons would lead us to one particular decision, but we have a strong "gut-feeling" that makes it difficult to go ahead with that action. Matters of love, family etc tend to be spoken of as "matters of the heart". So the figurative heart is part of our "soul".
Certain things appeal to our heart or our mind. Some things appeal to both. Other things are said to appeal to our more physical needs and desires. For example, a factual book might appeal to our mind and our desire for learning, whereas an emotional account of someone's life might touch our heart. Contrasting the heart and mind is quite common. In your example that contrasts the heart and soul, I think there might be a suggestion that touching the heart in isolation might be a little shallow. The comparison is the animated films of Disney and Ghibli. Saying that Disney films touch the heart, but implying that they do not touch the soul might be a suggestion that they are mawkish - deliberately sentimental in a shallow, manufactured way, perhaps? Saying that Ghibli films touch the soul seems to be a suggestion that they move people on a much deeper level.