Importance of Overall Reaction Order
This answer is mostly a compilation of the excellent comments by Spontification and Chet Miller.

Differential rate law

The overall reaction order tells you what happens to the rate when you change all concentrations by the same factor at the same time. For example, if you decrease all concentrations by a factor 2 and the overall order is 1, the reaction will be slower by a factor of 2. If the order is 2, slower by a factor of $2^2 = 4$, if the order is 3, slower by a factor of $2^3 = 8$ and so on (overall reaction orders higher than 3 are unusual for elementary reactions). Dilution by a factor 2 would be one way of lowering concentrations; however, not all species will decrease by a factor of 2 (solvent, hydronium concentration in a buffer, any other species buffered by a fast equilibrium).

Integrated rate law

Integrated rate laws describe how the rate changes (decreases) over time. All species concentrations that are part of the differential rate law will change over time unless they are catalysts. If one species concentration is very high compared to others (e.g. the solvent) then its change will be negligible. So the integrated rate law depends on more than the differential rate law.

However, if all of the species in the differential rate law are reactants, and these reactants are present at stoichiometric ratios, the overall reaction order is directly related to the integrated rate law (e.g. overall second order reaction gives second-order integrated rate law). If not, you can get, for example, a pseudo-first order reaction even though the overall order of reaction is two.