Both types of programs will have predominantly undergraduate / master's level mathematics majors, both programs' students will likely take courses in measure-theoretic probability theory (so that knowledge of introductory real and complex analysis are prerequisites), mathematical statistics, develop programming backgrounds, and learn some differential equations, e.g., ODEs, PDEs, SDEs.
Both types of PhD students are well aware of the (current?) opportunities in data science, machine learning, etc, and are giving these areas of study some considerable emphasis.
Those are the similarities. What are the differences?
It seems that people apply to the Phd in Stats as a way to avoid taking the Math subject GRE test, or that perhaps it is "easier" to get into a good PhD in Stats program, from having an undergrad math background.