Where is $MA$ more relevant than $\exists BPP$?
$\def\mr{\mathrm}$First, standard derandomization assumptions (the existence of a language in $\mr E$ that requires circuit size $2^{\Omega(n)}$) imply $\mr{promise\text-BPP=promise\text-P}$, hence
$$\mr{MA=\exists BPP=NP},$$
thus any difference between the two classes is likely just an artifact of our incomplete knowledge.
Having said that, there are several results in complexity theory involving $\mr{MA}$ that are not known to hold for $\exists\mr{BPP}$. In particular, there is a group of results of the form
$$\begin{align*}
\mr{PP\subseteq P/poly}&amp;\implies\mr{PP=MA},\\
\mr{PSPACE\subseteq P/poly}&amp;\implies\mr{PSPACE=MA},\\
\mr{EXP\subseteq P/poly}&amp;\implies\mr{EXP=MA},\\
\mr{NEXP\subseteq P/poly}&amp;\implies\mr{NEXP=MA}.
\end{align*}$$
(The first three results are due to Babai, Fortnow &amp; Lund, the last one is due to Impagliazzo, Kabanets &amp; Wigderson.)
If you allow me larger time bounds, an often quoted related result is that
$$\mr{MA\text-EXP\nsubseteq P/poly}$$
(here, $\mr{MA\text-EXP=MA\text-TIME}(2^{n^{O(1)}})$ is the exponential analogue of $\mr{MA}$). In fact, this result can be improved, nevertheless it is not known to be true for $\mr{NEXP^{BPP}}$ (which is the exponential analogue of $\exists\mr{BPP}=\mr{NP^{BPP}}$).