To answer the question: essentially, there's no difference between the two approaches. However, enum construct provides you with some additional supporting methods like values(), valueOf(), etc. which you'd have to write on your own with the class-with-private-constructor approach.
But yeah, I like how Java enums are mostly just like any other classes in Java, they can have fields, behaviors, etc. But to me what separates enums from the plain classes is the idea that enums are classes/types whose instances/members are predetermined. Unlike usual classes where you can create any number of instances from, enums only limit creation to known instances. Yes, as you've illustrated, you can also do this with classes with private constructors, but enums just make this more intuitive.
