Rate Expressions - A way of measuring the rate.
Rate Law - A way of predicting the rate given conditions (reactant concentrations).
If we wanted to know how fast a rxn was proceeding we could measure how quickly the concentration of any product was increasing over time or how quickly the concentration of any reactant was decreasing over time and use that to express the rate of the rxn (hence the name rate expression).
But a rate law shows how the rate depends upon the concentrations of the reactants so that you can make a prediction of the rate based upon starting conditions (if we start out with this much of this reactant and that much of that reactant, then what will be the rate of the rxn.
You can determine the rate expressions from the balanced chemical equation always.
You can determine the rate law from expirimental data always. But if it's an elementary rxn then (and only then) can you also determine the rate law from the balance rxn.
Hope this helps!