There is no performance difference.
A const_iterator is an iterator that points to const value (like a const T* pointer); dereferencing it returns a reference to a constant value (const T&) and prevents modification of the referenced value: it enforces const-correctness.
When you have a const reference to the container, you can only get a const_iterator.
Edited: I mentionned “The const_iterator returns constant pointers” which is not accurate, thanks to Brandon for pointing it out.
Edit: For COW objects, getting a non-const iterator (or dereferencing it) will probably trigger the copy. (Some obsolete and now disallowed implementations of std::string use COW.)