So, so many ways in which this is defective compared to actual sum types, some of them were already listed, but to me the most crucial, even if mostly about theory rather than practice, is valueless_by_exception.

The choice to provide exceptions everywhere as a error handling means C++ is obliged to admit that your std::variant may not have a value at all. Which blows up all of your type safety. In Rust I can say that this Pet is either a Dog or a Cat, and it cannot be neither, but in C++ std::variant of a Cat and a Dog might nevertheless be valueless_by_exception anyway. What can you do about that? I guess you could throw an exception...

In Rust if X is either A or B, and Y is either C or D, and Z is either E or F, then a structure with X, Y and Z has only eight possible states. In C++ this structure has 27 possible states because of exceptions.