First off, it is easier to start by answering your question in the uni variate case because then both estimators have an explicit solution as a function of the series of sorted observations $x_{(1)}\leq x_{(2)}\leq,...,x_{(n-1)}\leq x_{(n)}$.
The uni-variate version of the MVE is also known as the short estimator
and is the solution to
$\underset{1\leq i\leq(n-h+1)}{\arg.\min.}\;\;\;x_{(i+h-1)}-x_{(i)}\;\;\;[1]$
The uni-variate version of the MCD is also known as the truncated likelihood estimator and is the solution to:
$\underset{1\leq i\leq(n-h+1)}{\arg.\min.}\;\;\;\displaystyle\frac{1}{h}\displaystyle\sum_{j=1}^{i+h-1}x_{(j)}^2-\left(\frac{1}{h}\displaystyle\sum_{j=1}^{i+h-1}x_{(j)}\right)^2\;\;\;[2]$
Now, for symmetrical distribution, the functional corresponding to $[2]$ is proportional to that corresponding to $[1]$ (the likelihood of elliptical distribution computed over the $h$ most central observations).
You will find more details on this in this very clear paper:
- Maxbias Curves of Robust Location Estimators based
on Subranges, Croux and Haesbroeck, Metrika, 2001
So these estimators are solutions to different problems, with the definitions above it is easy to show that the solution to equation [1] does in general not equal the solution to equation [2].
Turning to the multivariate setting now (when $p$, the number of variables is greater than 1), the MVE looks for an ellipse through $p+1$ data points that contains $h$ data points, with smallest volume (this is just the multivariate generalization to elliptical distributions of the range for uni-modal distribution on the line). The MCD looks for the ellipse that contains $h$ data points and that has the smallest volume. By analogy to the univariate setting, in the case of the MCD, the ellipse does not in general pass through $p+1$ data points (just as the mean does not in general correspond to any observed value in the sample).
Note that these are for the estimators (to which the O.P. specifically refers), not about the differences between the respective algorithms (FastMVE and FastMCD).