Why is elasticity not defined simply as the slope of the graph? It makes more intuitive sense to me: as I increase the quantity, this is how much the price changes. Why is a much more complex formula used that takes into consideration the current quantity and price at a given point on the curve?
A key feature of elasticity is that it is a unit-free measure.  Suppose we are considering the price-elasticity of demand for apples which is $\frac{{\triangle}y/y}{{\triangle}x/x}$ where $y$ is quantity demanded and $x$ is price.  The elasticity will be the same whether apples are measured in pounds (lb) or kilograms (kg) because both $\triangle y$ and $y$ will be in the same units.  Similarly it will not matter in which currency price is measured, because $\triangle x$ and $x$ will be in the same currency units.  This makes comparisons of elasticities, between goods measured in different units, or between countries with different currencies, much more meaningful than comparisons of slopes.