Let $a$ and $b$ be positive real numbers such that $a^3 + b^3 = a + b.$  Simplify
\[\frac{a}{b} + \frac{b}{a} - \frac{1}{ab}.\]
This problem involves some algebra and fractions, so I want to simplify as much as possible. I notice that the given equation $a^3 + b^3 = a + b$ has a cubic term on one side and a linear term on the other, so I wonder if I can factor it somehow. One way to factor a sum of cubes is to use the formula $x^3 + y^3 = (x + y)(x^2 - xy + y^2).$ If I apply this formula to the given equation, I get $(a + b)(a^2 - ab + b^2) = a + b.$ Since $a$ and $b$ are positive, I can divide both sides by $a + b$ without changing the equation. This gives me $a^2 - ab + b^2 = 1.$ Now I have a simpler equation that relates $a$ and $b$, and I can use it to simplify the expression I want to find. The expression I want to simplify is $\frac{a}{b} + \frac{b}{a} - \frac{1}{ab}.$ I can rewrite this as $\frac{a^2 + b^2}{ab} - \frac{1}{ab}.$ Then I can combine the fractions and get $\frac{a^2 + b^2 - 1}{ab}.$ But I know from the equation I found earlier that $a^2 + b^2 - 1 = ab.$ So I can substitute this into the expression and get $\frac{ab}{ab}.$ This simplifies to $1.$ 