Here are some things you need to do:
Use \textbf{\Large\strut ...} in one of the header entries. This inserts a vertical strut of the appropriate size to obtain better spacing. There's also more ideas in terms of vertical adjustment of table entries in Column and row padding in tables.
Include the array package in your preamble
\usepackage{array}% http://ctan.org/pkg/array
and use a tabularx column specification of the form
\begin{tabularx}{\textwidth}{|>{\raggedright}X|>{\raggedleft\arraybackslash}X|}
% ...
\end{tabularx}
This inserts \raggedright [\raggedleft] in the first [second] column so items should be left- [right-]aligned. Note that lists do not conform to this adjustment. To that end, I've used varwidth's varwidth environment for capturing the list in a box of natural width (if it is smaller than the width of the column, or \linewidth).
Also, use \noindent before tabularx:
\noindent
\begin{tabularx}{\textwidth}{...}
%...
\end{tabularx}
otherwise your tabularx position won't sit flush with the margins.
Here is a complete MWE:

\documentclass{article}
\usepackage{array,tabularx,varwidth}% http://ctan.org/pkg/{array,tabularx,varwidth}
\begin{document}
\noindent
\begin{tabularx}{\textwidth}{|>{\raggedright}X|>{\raggedleft\arraybackslash}X|}
\hline
\textbf{{\Large\strut Activa}} & \textbf{{\Large Passiva}} \\
\hline\hline
\textbf{Aanwendingsvormen} van het vermogen & \textbf{Vermogensbronnen} \\[1.5\normalbaselineskip]
\begin{varwidth}[t]{\linewidth}
\begin{itemize}
\item Bezittingen
\item Vorderingen
\end{itemize}
\end{varwidth} &
\begin{varwidth}[t]{\linewidth}
\begin{itemize}
\item Eigen vermogen
\item Vreemd vermogen
\end{itemize}
\end{varwidth} \\[2\normalbaselineskip]
\textbf{Ondernemingsmiddelen} & \textbf{Ondernemingsbronnen} \\
Welke middelen heeft de onderneming om haar activiteit te ontwikkelen? &
Hoe financiert de onderneming de ondernemingsmiddelen.\\
\hline\hline
\textbf{{\Large\strut Vermogen} van de onderneming} &
\textbf{{\Large Vermogen} van de onderneming} \\
\hline
\end{tabularx}
\end{document}
Unknown (or incorrect) hyphenation patterns could also be problematic when it comes to line-breaking (not the case/issue here). If need be, you can manually force a different hyphenation using a blah\-blah\-blah approach.
varwidthto align theitemize. It's one option. Check the revision history for a complete picture of the changes. Since you're using atabularxenvironment for "parallel typesetting", the vertical adjustments around theitemizeis natural. The problems around these vertical adjustments stem from the boxing; for more on this discussion, see How to keep a constant\baselineskipwhen usingminipages (or\parboxes)?. – Werner Aug 22 '13 at 20:33sudo tlmgr install varwidth? Perhaps see How do I update my TeX distribution? or How do I install an individual package on a Linux system? – Werner Aug 23 '13 at 11:33