Let $ x $ be the probability we are looking for and $ y $ be the probability that they both spin the same number. By symmetry, it's clear that the probability of Zack getting a larger number than Max does is also equal to $ x $. Furthermore, all possible outcomes can be divided into three categories: Max gets a larger number than Zack does, Max and Zack get the same number, or Zack gets a larger number than Max. The sum of the probabilities of these three events is 1, which gives us the equation $ x+y+x=1 $. We can calculate $ y $ with a little bit of casework. There are four ways in which they can both get the same number: if they both get 1's, both get 2's, both get 3's or both get 4's. The probability of getting a 1 is $ \dfrac{1}{2} $, so the probability that they will both spin a 1 is $ \left(\dfrac{1}{2}\right)^2=\dfrac{1}{4} $. Similarly, the probability of getting a 2 is $ \dfrac{1}{4} $, so the probability that they will both spin a 2 is $ \left(\dfrac{1}{4}\right)^2=\dfrac{1}{16} $. The probability of getting a 3 is $ \dfrac{1}{6} $, so the probability that they will both get a 3 is $ \left(\dfrac{1}{6}\right)^2=\dfrac{1}{36} $ and the probability that they will both get a 4 is $ \left(\dfrac{1}{12}\right)^2=\dfrac{1}{144} $. This gives us $$y=\dfrac{1}{4}+\dfrac{1}{16}+\dfrac{1}{36}+\dfrac{1}{144}=\dfrac{25}{72}.$$Substituting this into $ 2x+y=1 $ gives us $ 2x=\dfrac{47}{72} $, so $ x=\boxed{\dfrac{47}{144}} $.