How is "interpretation" used differently in propositional vs. first-order logic?

In both propositional and predicate logic, the truth value of a formula is always either true or false, once an interpretation has been given. The set $\{true, false\}$ is not something you choose; it is a fixed part of how the logic works.

However, in predicate logic, formulas are not everything there is. Predicate logic also has terms, which are expressions that can be the arguments of relation symbols. (For example, in the language of arithmetic $2>3$ or $5=x+2$ are formulas; $2\cdot 3$ or $x+2$ are terms).

An interpretation in predicate logic tells you

• A set that the value of terms can be drawn from. (This is implicitly also the set that variables have their values in).
• An interpretation of each of the function symbols in the logical language. (For example, $+$ in the language of arithmetic).
• An interpretation of each of the predicate symbols -- that is a set of ordered tuples of values that make the predicate true when given as argument.

In propositional logic there are no terms, no functions, and predicates. All of the atomic formulas are propositional letters. Seen from the predicate-logic end we can view a propositional letter as a "predicate symbol" that takes no operands. Thus, if we apply the above sense of interpretation, such as symbol should be represented either by the set $\{()\}$ that contains the (unique) tuple of length 0, or by the empty set.

But this corresponds to a choice of whether the propositional letter is true or false -- so an "interpretation" for propositional logic is effectively the same as a map from the propositional letters to $\{true, false\}$. All we need to do is to write $true$ and $false$ instead of $\{()\}$ and $\varnothing$.

Since there are no terms, there is no need for an interpretation to specify which kind of values the terms would have if there were any.