Let's use the following gaseous chemical reaction as an example:
$$\ce{H2 + I2 -> 2HI}$$
When we study reaction kinetics, we oftentimes can't observe the changes in concentration of each reactant and product. Therefore the rate expression helps us to relate the rates of appearance/disappearance of the species present:
$$-\frac{1}{1}\frac{\Delta \left[ \ce{H2} \right]}{\Delta t}\;
= \; -\frac{1}{1}\frac{\Delta \left[ \ce{I2} \right]}{\Delta t}\;
=\; \frac{1}{2}\frac{\Delta \left[ \ce{HI} \right]}{\Delta t}\tag{rate expression}\label{rateex}$$
Note that each rate is modified by the inverse of the stoichiometric coefficient. I explicitly put $\frac{1}{1}$ in the expression just to emphasize this point. We can see that multiplying the coefficients of the reaction will not change the rate expression in the following example: let's multiply the coefficients in the chemical reaction by 2 and then write the corresponding rate expression.
$$\ce{2H2 + 2I2 -> 4HI}$$
$$-\frac{1}{2}\frac{\Delta \left[ \ce{H2} \right]}{\Delta t}\;
= \; -\frac{1}{2}\frac{\Delta \left[ \ce{I2} \right]}{\Delta t}\;
=\; \frac{1}{4}\frac{\Delta \left[ \ce{HI} \right]}{\Delta t}\tag{new expression}\label{newex}$$
The $\ref{rateex}$ and $\ref{newex}$ are numerically equivalent.
Full disclosure here: the term rate expression is ambiguous and is also used as a synonym for the rate law. I've also seen the rate expression referred to as the rate relationship.
The rate law or differential rate law relates the rate of a reaction to the concentration (or pressure) of the reactants. The rate of a reaction is proportional to the concentration (or pressure) of the reactants modified by some experimentally determined number called the reaction order.
$$\begin{align}
-\frac{\Delta\left[ \ce{H2} \right]}{\Delta t}
&\propto\left[\ce{H2}\right]^m\left[\ce{I2}\right]^n\\
-\frac{\Delta\left[ \ce{H2} \right]}{\Delta t}
&= k\left[\ce{H2}\right]^m\left[\ce{I2}\right]^n\tag{rate law}
\end{align}$$
Here, $m$ and $n$ are the reaction orders for $\ce{H2}$ and $\ce{I2}$, respectively, and the sum, $m + n$ yields the overall order of the reaction. The proportionality $a\propto b$ can be replaced with an equality $a = k b$. In kinetics, this constant of proportionality is referred to as the rate constant.
In summary: the rate expression tells us how the appearance/disappearance rates of the products and reactants relate to one another; the rate law tells us how the rate is related to the concentrations (or pressures) of reactants and the rate constant is a constant of proportionality that comes out of the rate law.
This answer is a summary of information found at the boundless website and from Tro's most recent general chemistry textbook.
