bullet in table not aligned

use

\newcommand\tabitem{\makebox[1em][r]{\textbullet~}}

This answer is similar in spirit to the one by @daleif. The difference is that I use the macro \RaggedRight (from the ragged2e package) instead of the basic \raggedright directive; the difference is that \RaggedRight permits hyphenation. I also use a \newcolumntype instruction to set up a new column type called "P" that sets in contents automatically in Ragged-Right paragraph mode.

\documentclass{beamer}
\usepackage{booktabs,array,ragged2e}
\newcolumntype{P}[1]{>{\RaggedRight\arraybackslash}p{#1}}
\newcommand{\tabitem}{\textbullet~~}
\begin{document}

\begin{frame}{Symbolic Math Toolbox}
\begin{itemize}
\item Don't do nasty calculations by hand!
\item Symbolics vs.\ Numerics
\end{itemize}
\begin{tabular}{| p{1.4cm} | P{4cm} | P{4cm} |} \hline    
& \textcolor{green}{Advantages} 
& \textcolor{red}{Disadvantages} \\ \hline
Symbolic 
& \tabitem Analytical solutions 
& \tabitem Sometimes can't be solved \\
& \tabitem Lets you intuit things about solution form 
& \tabitem Can be overly complicated \\ \hline 
Numeric 
& \tabitem Always get a solution 
& \tabitem Hard to extract a deep understanding \\
& \tabitem Can make solutions accurate 
& \tabitem Num.\ methods sometimes fail \\  
& \tabitem Easy to code 
& \tabitem Can take a while to compute \\  \hline
\end{tabular}
\end{frame}
\end{document}

Presumably because ~ is stretchable and your columns are attempting to to keep a straight right edge.

Probably better to use

\documentclass{beamer}
\usepackage{booktabs,array}% http://ctan.org/pkg/booktabs
\newcommand{\tabitem}{~~\llap{\textbullet}~~}
\begin{document}

\begin{frame}{Symbolic Math Toolbox}
\begin{itemize}
\item Don't do nasty calculations by hand!
\item Symbolics vs. Numerics
\end{itemize}
\begin{tabular}{| >{\raggedright}p{1.3cm} | >{\raggedright}p{4cm} | 
    >{\raggedright\arraybackslash}p{4cm} |} \hline &
  \textcolor{green}{Advantages} & \textcolor{red}{Disadvantages}
  \\
  \hline Symbolic & \tabitem Analytical solutions & \tabitem Sometimes
  can't be solved
  \\
  & \tabitem Lets you intuit things about solution form & \tabitem Can
  be overly complicated
  \\
  \hline Numeric & \tabitem Always get a solution & \tabitem Hard to
  extract a deep understanding
  \\
  & \tabitem Can make solutions accurate & \tabitem Num. methods
  sometimes fail
  \\
  & \tabitem Easy to code & \tabitem Can take a while to compute
  \\
  \hline
\end{tabular}
\end{frame}
\end{document}

and thus setting the cols ragged right. Note the \arraybackslash added to the last column.