The minimal model, when it exists, is the smallest transitive model of ZFC. This model will have the form $L_\alpha$ for some countable ordinal $\alpha$.
Since it is a model of $\text{ZFC}+V=L$, one gets on board all of the consequences of that theory in the minimal model. But there are many further properties true in the minimal model, when it exists, that are not provable from $\text{ZFC}+V=L$.
The minimal model satisfies $\text{Con}(\text{ZFC})$ and $\text{Con}(\text{ZFC}+\text{Con}(\text{ZFC}))$ and so forth, iterated many times, since these are arithmetic consequences of the existence of a transtive model of ZFC. These assertions go beyond what is provable in $\text{ZFC}+V=L$ alone, and so they provide examples of interesting assertions provably true in the minimal model but not provable in that theory.
Meanwhile, the minimal model satisfies the mildly self-referential assertion "the minimal model of ZFC does not exist". It can't exist inside itself. Consequently, the minimal model thinks there is no transitive model of ZFC.
The minimal model is pointwise definable—every individual is definable without parameters. This is because being a model of $V=L$ it has a definable global well order, and so the definable elements of it will form an elementary substructure, which will Mostowski collapse to some $L_\alpha$, but by minimality the collapse cannot be smaller than minimal model itself, and so it is isomorphic to its definable hull. So everything is definable without parameters.
The minimal model thus forms a counterexample ZFC universe for the Math tea argument, which attempts to show that there must be nondefinable real numbers, since there are only countably many definitions in set theory, but uncountably many real numbers.
The minimal model has no inaccessible cardinals, nor any worldly cardinals. If it did, then they would provide smaller transitive models.
The consistency strength of the existence of the minimal model is strictly stronger than ZFC, but weaker than the existence of a worldly cardinal, which is itself weaker than the existence of an inaccessible cardinal.