I'm going to add to the existing answers because modern C++ is now a thing and official Core Guidelines have been created to help with questions such as these.
Here's a relevant section from the guidelines:
C.2: Use class if the class has an invariant; use struct if the data members can vary independently
An invariant is a logical condition for the members of an object that a constructor must establish for the public member functions to assume. After the invariant is established (typically by a constructor) every member function can be called for the object. An invariant can be stated informally (e.g., in a comment) or more formally using Expects.
If all data members can vary independently of each other, no invariant is possible.
If a class has any private data, a user cannot completely initialize an object without the use of a constructor. Hence, the class definer will provide a constructor and must specify its meaning. This effectively means the definer need to define an invariant.
Enforcement
Look for structs with all data private and classes with public members.
The code examples given:
struct Pair { // the members can vary independently
string name;
int volume;
};
// but
class Date {
public:
// validate that {yy, mm, dd} is a valid date and initialize
Date(int yy, Month mm, char dd);
// ...
private:
int y;
Month m;
char d; // day
};
Classes work well for members that are, for example, derived from each other or interrelated. They can also help with sanity checking upon instantiation. Structs work well for having "bags of data", where nothing special is really going on but the members logically make sense being grouped together.
From this, it makes sense that classes exist to support encapsulation and other related coding concepts, that structs are simply not very useful for.