Great question! It turns out that most reactions are more complicated than they appear. Let's look at an example to understand why:

This reaction describes "water splitting," where water is decomposed into the gases hydrogen and oxygen. The way the reaction is written, one might think that two water molecules smash into each other, and out pops an O₂ molecule and two H₂ molecules. This is NOT actually what happens at the atomic level. There are a lot of intermediate "elementary steps" that occur between the reactants (water) and the products (oxygen and hydrogen). The list of elementary steps looks something like:

(1) ( H₂0 <-> OH + H ) x₂ (happens 2 times separately).

These steps usually occur on the surface of a metal catalyst, which makes it possible to form all the intermediate species, like a single H atom, which are usually unstable on their own. If one studies each elementary step, the reaction rate exponent is equal to the stoichiometry coefficient. For example the forward reaction rate for reaction 5 is:

r₅ = k₅ * [H]²

and the backward reaction rate is:
r_(-5) = k_(-5) * [H₂]

where k₅ and k_(-5) are rate constants, [H] is the concentration of H atoms, and [H₂] is the concentration of diatomic H (molecule).

When we look at the overall reaction:
H₂O <-> 2 H₂ + O₂
and try to write a forward rate expression:
r__tot = k__tot * [H₂O]^x

the rate constant, k__tot, and exponent, x, are going to have complex dependencies on the rate constants and exponents of each of the elementary steps. It is very hard to calculate what the exponent, x, will be from the elementary steps, so usually it is just measured.

The reaction quotient on the other hand is an equilibrium quantity. This means it does not care about the path that the atoms take to get from reactants to products. Since the actually mechanism doesn't matter for the equilibrium quotient, the stoichiometry coefficients tell us everything we need to know about how the reactants and products are related.

In short, it is because the reaction quotient is a quantity which describes the state of the system once everything has “settled down,” or is in equilibrium. Since it only depends on the final concentrations of each species and not the mechanism by which those species formed, we can define it in terms of stoichiometric coefficients. This because stoichiometry dictates the ratios of elements which can be formed by the reaction. On the other hand reaction rates are very much dependent on the mechanism, or reaction pathway. That is why they are experimentally determined and can NOT be determined by stoichiometry. For what is called an “elementary reaction” the reaction rate can be written using stoichiometric coefficients as exponents. The textbook definition of an elementary reaction is one that proceeds exactly as written in the stoichiometric equation. In reality it is more complicated, but that’s another story!