Output forms provide textual representations of Wolfram Language expressions. In some cases these textual representations are also suitable for input to the Wolfram Language. But in other cases they are intended just to be looked at, or to be exported to other programs, rather than to be used as input to the Wolfram Language.
TraditionalForm uses a large collection of ad hoc rules to produce an approximation to traditional mathematical notation:
Output forms not normally used for Wolfram Language input.
"Low-Level Input and Output Rules" discusses how you can create your own output forms. You should realize however that in communicating with external programs it is often better to use the Wolfram Symbolic Transfer Protocol (WSTP) to send expressions directly than to generate a textual representation for these expressions.
■ Exchange textual representations of expressions.
■ Exchange expressions directly via WSTP.
Two ways to communicate between the Wolfram System and other programs.