Role of Term Constants in Simply Typed Lambda Calculus
As per Sam's request, I rephrase my comment as an answer.

Constants play an important role in λ-calculi (not just the simply typed variant).


They are often convenient: even though we can usually represent our target data using
$\lambda$-terms without constants this tends to be unwieldy. For example
integers 
directly by some variant of an unary encoding, e.g. the Church encoding has $$\lambda g x. \underbrace{g ( g ( ... g ( g\ x ) .. ) ) }_{n}$$ as representation of $n$. That leads to cumbersome, unreadable terms. With constants we can  do arithmetic with 0,1,2,..., and +,−,∗,... 
In a typed calculus, we need  constructors for types, and constants play this rule. E.g. 0,1,2,..., or +,−,∗,...
for the type of integers, true,false... or ∨,∧ for booleans etc. 


Note that the STLC was born with constants : Church's original
formulation contains constants $N_{oo}$, $A_{ooo}$, $\Pi_{o(o\alpha)}$
as well as $\iota_{\alpha(o\alpha)}$, representing negation,
disjunction, universal quantification and the selection operator,
respectively.