I'm currently looking at The Story of a Mother as part of the Future Learning Hans Christen Andersen course and its depth is amazing. It tackles the theme of grief and suffering, not a common subject for a fairytale. Nor is it an easy theme to write about without becoming overly sentimental or harshly cold.
I recommend HCA's story as I think he gets the balance right and the mother is a well rounded, sympathetic character. It's definitely not a story you'd read at bedtimes to younger children. There has to be a certain level of maturity to pick up on the themes of this story. I'd not come across this story before the course but am glad I have. HCA wrote with depth, understanding and sympathy and I'm appreciating his skills as a writer more. His ability to write stories which reach out to children and adults is amazing.