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.
Short answer: they are two different (but overlapping) subjects.
Medium answer: Applied math is a broad term that can mean anything from mathematical physics to machine learning to cryptography to numerical analysis to biostatistics to...the list goes on. And it can mean different things at different universities/departments. There is plenty of applied math that doesn't involve statistics. I would go as far as to say that less than half of applied math involves statistics, although the prevalence of statistics is growing. And even though, philosophically speaking, statistics may be a form of applied mathematics, there are plenty of subjects within statistics that are not what people usually refer to as applied math.