\TeX and \LaTeX differently expl3-x-written (exhaustive expansion) to an auxiliary file

You don't want \exp_not:V, but rather \exp_not:N

\documentclass{article}
\usepackage{xparse}

\ExplSyntaxOn

\iow_new:N \g_output_stream

\iow_open:Nn \g_output_stream { \c_sys_jobname_str.sbj }
\iow_now:Nn \g_output_stream { \LaTeX{} }
\iow_now:Nn \g_output_stream { \TeX{} }
\iow_now:Nx \g_output_stream { \exp_not:N \LaTeX{} }
\iow_now:Nx \g_output_stream { \exp_not:N \TeX{} }
\iow_close:N \g_output_stream

\ExplSyntaxOff

\begin{document}
\end{document}

The auxiliary file will have

\LaTeX {}
\TeX {}
\LaTeX {}
\TeX {}

What's the difference? In this case \exp_not:V does a one step expansion, which is not desired here, because \LaTeX and \TeX are not variables, but parameterless functions. Recall that V stands for variable and is used to get its value, not subject to further expansion.

If we do \show\TeX, the result is

\TeX=macro:
->T\kern -.1667em\lower .5ex\hbox {E}\kern -.125emX\@.

On the other hand, the result of \show\LaTeX is

   > \LaTeX=macro:
   ->\protect \LaTeX  .

Now \LaTeX is defined in latex.ltx using \DeclareRobustCommand, and as @egreg has pointed out, \MakeRobust can protect an existing macro in the same way.

\documentclass{article}
\MakeRobust{\TeX}

\usepackage{xparse}

\ExplSyntaxOn

\iow_new:N \g_output_stream

\iow_open:Nn \g_output_stream { \c_sys_jobname_str.sbj }
\iow_now:Nn \g_output_stream { \LaTeX{} }
\iow_now:Nn \g_output_stream { \TeX{} }
\iow_now:Nx \g_output_stream { \exp_not:V\LaTeX{} }
\iow_now:Nx \g_output_stream { \exp_not:V\TeX{} }
\iow_close:N \g_output_stream

\ExplSyntaxOff

\begin{document}
\end{document}