As Tal Pressman answered at When should you use a class vs a struct in C++?:
From the C++ FAQ:
The members and base classes of a struct are public by default, while in class, they default to private. Note: you should make your base classes explicitly public, private, or protected, rather than relying on the defaults.
struct and class are otherwise functionally equivalent.
OK, enough of that squeaky clean techno talk. Emotionally, most developers make a strong distinction between a class and a struct. A struct simply feels like an open pile of bits with very little in the way of encapsulation or functionality. A class feels like a living and responsible member of society with intelligent services, a strong encapsulation barrier, and a well defined interface. Since that's the connotation most people already have, you should probably use the struct keyword if you have a class that has very few methods and has public data (such things do exist in well designed systems!), but otherwise you should probably use the class keyword.
http://www.parashift.com/c++-faq-lite/classes-and-objects.html#faq-7.8
I think one addition to this reason could be that C already had structs. When Bjarne Stroustrup designed C++, he wanted to add classes with all the functionalities you listed in your original question, features which were not present in C. When implementing those features, he probably realised it didn't make sense to make two separate implementations for struct and class (except the public/private default visibility).
TL/DR: in C++ structs/classes describe the intent of the programmer to have POD or more complex abstractions, and the introduction of the class keyword is probably here for historical reasons (add an additional keyword in order to create featurefull classes, then backport those features into the struct keyword because it's more pragmatic to implement).