Change caption in table from Table 2 to Table 1A

I would do this using the subcaption package- the key parts of the MWE below are

\usepackage{subcaption}
\DeclareCaptionSubType*[Alph]{table}
\DeclareCaptionLabelFormat{mystyle}{Table~\bothIfFirst{#1}{ }#2}
\captionsetup[subtable]{labelformat=mystyle}

The output is

Note that I've put your 'sub' tables into a subtable environment, defined by the subcaption package.

\documentclass{article}

\usepackage{subcaption}
\DeclareCaptionSubType*[Alph]{table}
\DeclareCaptionLabelFormat{mystyle}{Table~\bothIfFirst{#1}{ ̃}#2}
\captionsetup[subtable]{labelformat=mystyle}


\begin{document}

\begin{table}[hbp!]
    \centering
    \caption{This caption describes contents of Table 1, gives the big picture.}
    \label{TestTable}
    \begin{subtable}{\textwidth}
            \centering
        \caption{Describe contents of Table 1.A}
        \begin{tabular}{|c|c|c|c|c|c|}
            \hline
            Recom Strength & Highest Idos Risk & 2nd Highest & Middle & 2nd Lowest & Lowest \\
            \hline
            1              & -0.41             & -0.07       & 1.27   & -0.78      & 0.73   \\
            \hline
            2              & -0.85             & -1.88       & 0.01   & 0.75       & 2.32   \\
            \hline
            3              & -0.48             & -1.0        & -0.56  & 0.47       & 1.01   \\
            \hline
            4              & 0.04              & 0.76        & 1.34   & 1.78       & -1.27  \\
            \hline
            5              & -1.35             & -1.52       & 0.04   & 1.23       & 0.51   \\
            \hline
        \end{tabular}
    \end{subtable}

    \begin{subtable}{\textwidth}
            \centering
        \caption{Describe contents of Table 1.B}
        \begin{tabular}{|c|c|c|c|c|c|} 
            \hline
            Recom Strength & Highest Idos Risk & 2nd Highest & Middle & 2nd Lowest & Lowest \\
            \hline
            1              & -0.41             & -0.07       & 1.27   & -0.78      & 0.73   \\
            \hline
            2              & -0.85             & -1.88       & 0.01   & 0.75       & 2.32   \\
            \hline
            3              & -0.48             & -1.0        & -0.56  & 0.47       & 1.01   \\
            \hline
            4              & 0.04              & 0.76        & 1.34   & 1.78       & -1.27  \\
            \hline
            5              & -1.35             & -1.52       & 0.04   & 1.23       & 0.51   \\
            \hline
        \end{tabular}
    \end{subtable}%
\end{table}

\end {document}

Unrelated to your question- a few folks would recommend the booktabs package, and would also recommend removing your vertical lines. I also noticed that these tables give you overfull hboxes- this should really be fixed before sending this to print.

A sans-caption solution that can be optimised (especially the needed \startsubcaptions).

Code

\documentclass[12pt]{article}
\usepackage{etoolbox}

\newcounter{subtable}
\renewcommand*{\thesubtable}{\Alph{subtable}}
\newcounter{subfigure}
\renewcommand*{\thesubfigure}{\Alph{subfigure}}

\makeatletter
\newcommand*{\startsubcaptions}{%
  \setcounter{sub\@captype}{0}
  \csappto{the\@captype}{.\csname thesub\@captype\endcsname}%
}
\newcommand*{\subcaption}[2][]{%
    \stepcounter{sub\@captype}%
    \addtocounter{\@captype}{-1}%
    \ifx\\#1\\\caption{#2}\else\caption[#1]{#2}\fi%
}
\newcommand*{\nonumbercaption}[1]{%
  \refstepcounter{\@captype}%
  {\centering #1\par}%
}
\makeatother
\begin{document}
\begin{table}[!hbp]
\nonumbercaption{This caption describes contents of Table 1, gives the big picture.}\label{table1}%
\startsubcaptions%
\subcaption{Describe contents of Table 1.A}\label{table1a}
\subcaption{Describe contents of Table 1.B}\label{table1b}
\end{table}

\ref{table1}, \ref{table1a}, \ref{table1b}

\begin{table}[!hbp]
\caption{This caption describes contents of Table 2, gives the big picture.}\label{table2}%
\startsubcaptions%
\subcaption{Describe contents of Table 2.A}\label{table2a}
\subcaption{Describe contents of Table 2.B}\label{table2b}
\end{table}

\ref{table2}, \ref{table2a}, \ref{table2b}
\end{document}

Output