Both function types contain a straight line on a plane, so if a student asks for the difference between them what would be a standard or recommended way to explain it?
A constant function does not vary with the independent variable: its value remains the same no matter what $x$ is (assuming $x$ is the independent variable).
As a formula, a line can be described by $y = ax + b$. The special case of a constant function has $a = 0$, so the formula can be written succinctly as $y = b$.
If you want to bring in the idea of slope, note that a constant function has slope zero and that the graph is a horizontal line.