How to properly use subcaption with threeparttable

I replaced the subcaption with the subfig package and used the \subfloat command, now it works like a charm with \tnotes in the (sub)caption.

Another option, without using threeparttable, can be found here using a simple \footnotesize inside figure environment.

\documentclass[a4paper]{memoir}
\usepackage{geometry,array,graphicx,float,caption}
\usepackage{subfig}
\usepackage{threeparttable}

\begin{document}
\captionsetup[subfloat]{position=bottom}
\begin{figure}[ht]
\caption{Samples picture} \label{fig:picture_temp}
    \centering
\begin{threeparttable}
        \subfloat[Before]{
        \includegraphics[width=6cm]{example-image-b}
        \label{fig:1a} 
        }
    ~
        \subfloat[After \tnote{a}]{
        \includegraphics[width=6cm]{example-image-c}
        \label{fig:1b}
        }    
    \begin{tablenotes}
     \item Diference in samples over the range of temperatures.
     \item[a] Calculated.
    \end{tablenotes}
    \end{threeparttable}
\end{figure}
\end{document}

Inside threeparttable the meaning of \@caption is changed. You want that inside of subfigure the meaning is as established by caption.

\documentclass[a4paper]{memoir}
\usepackage{geometry,array,graphicx,caption}
\usepackage{subcaption}
\usepackage{threeparttable}
\usepackage{etoolbox}

\makeatletter
\AtBeginEnvironment{subfigure}{\let\@caption\caption@original@caption}
\AtBeginDocument{\let\caption@original@caption\@caption}
\makeatother

\begin{document}

\begin{figure}[ht]
\caption{Samples picture} \label{fig:picture_temp}
    \centering
\begin{threeparttable}
    \begin{subfigure}{0.45\textwidth}
        \centering
        \includegraphics[width=6cm]{example-image-b}
        \caption{Before}
        \label{fig:1a}
    \end{subfigure}
    ~
    \begin{subfigure}{0.45\textwidth}
        \centering
        \includegraphics[width=6cm]{example-image-c}
        \caption{After}
        \label{fig:1b}
    \end{subfigure}    
    \begin{tablenotes}
     \item Difference in samples over the range of temperatures.
    \end{tablenotes}
    \end{threeparttable}
\end{figure}
\end{document}

The original meaning (as further modified by memoir) has to be saved at begin document. With \AtBeginDocument I restore that meaning, but only inside subfigure. You may need the same for tables.