If code is a set of symbols designed to convey meaning (or to facilitate communication), then language (as spoken or written) is a code, but there are codes that are non-linguistic. So not all codes are languages. Unless, of course, you consider every symbolic system designed to convey meaning to be a language. In that case, code is just another word for language.
However, it seems to me that code is generally used to mean a symbolic system that it takes special education to understand, while language is something people are built to understand. I.e., code is man-made, while language has evolved, but can not be considered to have been made.