What are the differences between a PhD in Statistics and a PhD in Applied Math?
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.