It depends on the context. For example, in physics we often represent constants with letters. For example, the equation for gravitational force between two masses ($m_1$and $m_2$) separated by a distance $r$ is given by $F = G(m_1m_2/r^2)$ where G is a constant.
But in mathematics we might write $y = ax^2 + bx + c$ where $a$, $b$, and $c$ are constants, but we call them coefficients.
So, what is the difference? The difference is that in the physics expression G stands for a known value, $6.6743 x 10^-11 m^3/kgs^2$, but in the math example, the coefficients are arbitrary and unspecified. They are constants in that they do not change like x and y, but arbitrary in that we haven't assigned a specific value to them.
In the sense of how they operate, coefficients are constants. Whether we call them constants or coefficients depends on whether we've actually actually assigned a value to them.