Different languages define the words "cast" and "convert" differently; I don't think the question is meaningful other than in reference to a particular language.
In C, for example, the term "cast" properly refers only to an explicit cast operator, consisting of a type name in parentheses preceding the expression to be converted. A "conversion" converts a value of one type to a value of another type; some conversions are implemented by re-interpreting the bits that make up the representation, but it's defined as a value-to-value conversion. (Yes, that's true even for pointer conversions; it's possible for different pointer types to have different representations.)
Note that there is no such thing as an "implicit cast" in C.
Some conversions are explicit, specified by a cast operator. Others are implicit, and are applied in certain cases when an expression of one type is used in a context that needs an expression of a different type. The conversion performed is exactly the same in either case.
double x = 1.23;
int y = (int)x; /* A cast, or explicit conversion, setting y to 1 */
int z = x; /* An implicit conversion, setting z to 1. */
C++ is similar; it has the same casts and conversions as C, and it adds a functional notation equivalent to a C-style cast expression, plus 4 more specific keywords: