Remove (a), (b) from subfigure numbering, but keep the subfigure caption?

You can use

\captionsetup[subfigure]{labelformat=empty}

locally:

\documentclass{article}
\usepackage{graphicx}
\usepackage{subcaption}

\begin{document}

\begin{figure}
\captionsetup[subfigure]{labelformat=empty}
\begin{subfigure}{.5\textwidth}
\centering
\includegraphics[height=3cm]{example-image-a}
\caption{Test subfigure 1}
\end{subfigure}%
\begin{subfigure}{.5\textwidth}
\centering
\includegraphics[height=3cm]{example-image-b}
\caption{Test subfigure 2}
\end{subfigure}%
\caption{Two subfigures}
\end{figure}

\end{document}

Another option is to use \caption* instead of \caption:

\documentclass{article}
\usepackage{graphicx}
\usepackage{subcaption}

\begin{document}

\begin{figure}
\begin{subfigure}{.5\textwidth}
\centering
\includegraphics[height=3cm]{example-image-a}
\caption*{Test subfigure 1}
\end{subfigure}%
\begin{subfigure}{.5\textwidth}
\centering
\includegraphics[height=3cm]{example-image-b}
\caption*{Test subfigure 2}
\end{subfigure}%
\caption{Two subfigures}
\end{figure}

\end{document}

Like subcaption, the subfig package also have the same option:

\documentclass{article}
\usepackage{graphicx,subfig}
\begin{document}
\captionsetup[subfigure]{labelformat=empty}
\begin{figure}
\centering
\subfloat[Test subfigure 1]{\includegraphics[width=.4\linewidth]{example-image-a}}
\hfill
\subfloat[Test subfigure 2]{\includegraphics[width=.4\linewidth]{example-image-b}}
\caption{Two subfigures}
\end{figure}
\end{document}

But if you do not need the numeration of the subfloats captions ... there are some reason to still use subcaption or subfig ?

For example, the same result can be obtained simply with 2 columns:

\documentclass{article}
\usepackage{graphicx,multicol}
\begin{document}
\begin{figure}
\begin{multicols}{2}
\centering
\includegraphics[width=.7\linewidth]{example-image-a}\\
Test subfigure 1
\includegraphics[width=.7\linewidth]{example-image-b}\\
Test subfigure 2
\end{multicols}
\caption{Two subfigures}
\end{figure}
\end{document}

Just stack the subcaption text under the subfigure. You can change the fontsize of the subcaptions just by changing the \footnotesize to the desired size. The gap between figure and caption is controlled by the optional argument to \stackunder and is, in this example, set to 5pt (the default is 3pt).

Note that the subcaptions in this technique, will not automatically word-wrap. In your case, there was no need for wrapping, since they were short labels. If wrapping were needed, the last argument to \stackunder could be a \parbox.

I further assumed, that since you wished to remove the (a), (b), that you had no desire to separately reference the individual subfigures. You can still reference the overall figure with \ref{}.

\documentclass{article}
\usepackage{stackengine}
\usepackage[demo]{graphicx}
\begin{document}
\begin{figure}
\footnotesize
\stackunder[5pt]{\includegraphics[width=2in,height=.7in]{fileA}}{MRI-CGCM3}%
\hspace{1cm}%
\stackunder[5pt]{\includegraphics[width=2in,height=.7in]{fileB}}{NorESM1-M}
\caption{November to April}
\end{figure}
\end{document}