= Solution
The <set> $\Sigma$ is <satisfiable> if there is one $L$-structure $\mathcal A$ satisfying every <first-order sentence> in $\Sigma$ simultaneously. It is finitely <satisfiable> if each finite <subset> $\Sigma_0\subseteq\Sigma$ has a <first-order model>; that <first-order model> is allowed to depend on the chosen finite <subset>.
Thus the two statements have different <quantifier orders>:
$$
\exists\mathcal A\ \forall\sigma\in\Sigma\ (\mathcal A\models\sigma),
\qquad
\forall\Sigma_0\subseteq_{\rm finite}\Sigma\ \exists\mathcal A\ (\mathcal A\models\Sigma_0).
$$
Satisfiability immediately implies finite satisfiability by using the same <first-order structure> for every finite <subset>. The converse is the substantive content of <compactness theorem>.
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