In modal logic, we have
$\Box(p\wedge q)\leftrightarrow\Box p\wedge\Box q$
but only
$\Box p\vee\Box q\rightarrow\Box(p\vee q)$,
where $\Box$ is the operator of necessity.
Do the same relations hold in infinitary modal logic, i.e. do we have
$\Box\bigwedge_{n\in\mathbb{N}}p_n\leftrightarrow\bigwedge_{n\in\mathbb{N}}\Box p_n$
but only
$\bigvee_{n\in\mathbb{N}}\Box p_n\rightarrow\Box\bigvee_{n\in\mathbb{N}}p_n$?
In general, when we think of modal logic we think of normal modal logic, where the principles you mention ($\Box (p\wedge q)\leftrightarrow \Box p\wedge \Box q$) are indeed valid. But notice that these are genuine axioms: there exist systems of modal logic where they are not valid. If you are used to Kripke semantics, such systems cannot be captured there, and you need to go to weaker semantics (e.g. neighbourhood semantics).
In the same vein, the infinitary laws you mention do not need to hold generally. But in Kripke semantics they do hold. In fact, regardless of how big you make the conjunctions and disjunctions, in Kripke semantics the box will distribute over the conjunction. Moreover, notice that since the other law already need not hold in the finite case, the infinite law is optional when working with the infinite case as well.
