Tables do not align side by side

Problems:

• Your tables exceed the text width, because you use two times 0.5\linewidth plus \hspace{0.5cm}. This can cause alignment problems and is problematic in general.

• You align minipages at the bottom, using [b], not at the top. I guess that's because of the caption, but this way you don't control the top.

I suggest:

• Always take care of the line width resp. text width. Note: also line breaks or blanks between minipages count and require space. Insert % before an end of the line to prevent that.

• Use the optional height argument of minipage or \parbox. You could use a fixed height, top alignment (below pos, inner-pos), perhaps \vfill to achieve alignment at top and bottom (captions).

  Syntax: \begin{minipage}[pos][height][inner-pos]{width}

• Further tip: vertical lines are really distracting. For me, some horizontal lines are usually fine. I strongly recommend to use the booktabs package. Just look into good books to see for yourself that fine tables don't interrupt rows by vertical lines.

Try this:

\documentclass{article}
\usepackage{tabularx}
\usepackage{multicol}

\begin{document}
\begin{table}[ht]
\begin{minipage}[b]{0.5\linewidth}%\centering
\begin{tabular}{|c|c|c|c|}
\hline
    & One & Twp & Three \\
\hline
1 & Subject A: & Subject A & Subject  \\
2 & Subject B & Subject B& Subject  \\
3 & Subject C & Subject C & Subject   \\
4 & Subject D & Subject D & Subject  \\
4 & Subject E & Subject E & Subject   \\
5 &Subject F & Subject F & Subject  \\
6 & Subject G & Subject G & Subject  \\
7 &Subject H &Subject H &Subject   \\
\hline
\end{tabular}
\caption{Table 1}
\end{minipage}
\hspace{0.5cm}
\begin{minipage}[b]{0.5\linewidth}
\centering
\begin{tabular}{|c|c|c|c|}
\hline
    & One & Twp & Three \\
 \hline
1 & Subject A: & Subject A & Subject  \\
2 & Subject B & Subject B& Subject  \\
3 & Subject C & Subject C & Subject   \\
4 & Subject D & Subject D & Subject  \\
4 & Subject E & Subject E & Subject   \\
5 &Subject F & Subject F & Subject  \\
6 & Subject G & Subject G & Subject  \\
7 &Subject H &Subject H &Subject   \\
\hline
\end{tabular}
\caption{Table 2}
\end{minipage}
\end{table}

\end{document}

This seems to work just fine.

\documentclass{article}
\usepackage{fullpage}
\begin{document}
\begin{table}
\noindent
\parbox{.5\textwidth}{%
\centering
\begin{tabular}{|c|c|c|c|}
\hline
    & One & Twp & Three \\
 \hline
1 & Subject A: & Subject A & Subject  \\
2 & Subject B & Subject B& Subject  \\
3 & Subject C & Subject C & Subject   \\
4 & Subject D & Subject D & Subject  \\
4 & Subject E & Subject E & Subject   \\
5 &Subject F & Subject F & Subject  \\
6 & Subject G & Subject G & Subject  \\
7 &Subject H &Subject H &Subject   \\
\hline
\end{tabular}
\caption{Table 1}
}%
\parbox{.5\textwidth}{%
\centering
\begin{tabular}{|c|c|c|c|}
\hline
    & One & Twp & Three \\
 \hline
1 & Subject A: & Subject A & Subject  \\
2 & Subject B & Subject B& Subject  \\
3 & Subject C & Subject C & Subject   \\
4 & Subject D & Subject D & Subject  \\
4 & Subject E & Subject E & Subject   \\
5 &Subject F & Subject F & Subject  \\
6 & Subject G & Subject G & Subject  \\
7 &Subject H &Subject H &Subject   \\
\hline
\end{tabular}
\caption{Table 2}
}%
\end{table}
\end{document}

This wasn't part of your question, but I think if you follow the advice given in the booktabs documentation, you can produce even better looking tables. Compare.

\documentclass{article}
\usepackage{fullpage}
\usepackage{booktabs}
\begin{document}
\begin{table}
\noindent
\parbox{.5\textwidth}{%
\centering
\begin{tabular}{rllll}
\toprule
    & \multicolumn{1}{c}{One}
    & \multicolumn{1}{c}{Two}
    & \multicolumn{1}{c}{Three} \\
\midrule
1 & Subject A: & Subject A & Subject  \\
2 & Subject B & Subject B& Subject  \\
3 & Subject C & Subject C & Subject   \\
4 & Subject D & Subject D & Subject  \\
4 & Subject E & Subject E & Subject   \\
5 &Subject F & Subject F & Subject  \\
6 & Subject G & Subject G & Subject  \\
7 &Subject H &Subject H &Subject   \\
\bottomrule
\end{tabular}
\caption{Table 1}
}%
\parbox{.5\textwidth}{%
\centering
\begin{tabular}{rllll}
\toprule
    & \multicolumn{1}{c}{One}
    & \multicolumn{1}{c}{Two}
    & \multicolumn{1}{c}{Three} \\
\midrule
1 & Subject A: & Subject A & Subject  \\
2 & Subject B & Subject B& Subject  \\
3 & Subject C & Subject C & Subject   \\
4 & Subject D & Subject D & Subject  \\
4 & Subject E & Subject E & Subject   \\
5 &Subject F & Subject F & Subject  \\
6 & Subject G & Subject G & Subject  \\
7 &Subject H &Subject H &Subject   \\
\bottomrule
\end{tabular}
\caption{Table 2}
}%
\end{table}
\end{document}