Difference between polynomial and power series

The formal polynomial expression is a function because you can always substitute a numerical value for $x$ and get a number. That's not necessarily the case for a power series, since there may well be values for $x$ for which the infinite series does not converge. When restricted to the values of $x$ for which it does converge, it defines a function.

That's why your instructor was careful to call the formal power series just an expression.

Related, possibly helpful: What actually is a polynomial?

It's an expression rather than a function because the series might not converge anywhere except at $x=0$.