What is the significance of Law of Multiple Proportions?
These laws all date back from the very early days of chemistry and are only taught at school to give pupils a way to grasp parts of the subject more easily. Except for the law of conservation of mass, which some professors/supervisors love stating frequently, they are hardly mentioned in universities outside of science history classes.

That said, as I understood them, they refer to slightly different cases. The law of definite proportions should be understood to read:


  No matter how much mass of two elements will react with each other, the product will contain the masses of its elements in a certain ratio and this ratio only.


Taking an easy example, if I reacted $46~\mathrm{g}$ of sodium and $16~\mathrm{g}$ of oxygen, I would get $62~\mathrm{g}$ of $\ce{Na2O}$ where the ratio of sodium to oxygen is $46:16$. If I had instead reacted $92~\mathrm{g}$ of sodium with an unlimited amount of oxygen, I would have gotten $124~\mathrm{g}$ of the same product which contains sodium and oxygen in a $46:16$ ratio. No matter how much sodium I let react with how much oxygen, the product will contain the definite proportion of sodium to oxygen $46:16$.*

The law of multiple proportions now states that some elements are special. For example, iron can react with chlorine to give two different chlorides: $\ce{FeCl2}$ and $\ce{FeCl3}$. For each of the two you can, as you realised set up a law of definite proportions which often gives you a rather weird rational ratio. However, when comparing those two with each other, you find that the mass of chlorine in $\ce{FeCl3}$ is 1.5 times as much as the mass of chlorine in $\ce{FeCl2}$ — corresponding to a $3:2$ ratio. And it is these very small (usually single digit) whole numbers that are actually meant.

Taking a different example, the relative atomic masses of sulphur and chlorine are $32.07$ and $35.45$, respectively. So a sulphur dichloride would have mass ratios of $32.07:70.9$ — far from any small number, but still a rational ratio. Sulphur tetrachloride exists, too, with another pretty unclean ratio. However, the ratio of the chloride contents is $1:2$ — isn’t that a nice simple ratio?



*: This example is not technically correct. Burning sodium gives $\ce{Na2O2}$, sodium peroxide. But I wanted nice and whole numbers and oxygen and sodium have nice and whole mass numbers.