Why do high tides vary month to month?
The dynamics of the tides are quite complex. The main idea is that gravity from the Moon and the Sun affect water (and everything else) on Earth. The issue is that there are several motions that alter the distance between the 3 systems and those motions cause interactions between the different frequencies involved. The Equilibrium Theory of Tides separates the different effects into a set of constituents by conducting a harmonic analysis. The relevant periods are:


the lunar day (period of lunar rotation), 24.84 mean solar hours.
the sidereal month (period of lunar declination), 27.32 mean solar days.      
the tropical year (period of solar declination), 365.24 mean solar days.
the period of the lunar perigee, 8.85 years (1 year = 365.2421988 days).
the period of the lunar node, 18.61 years.
the period of the solar perihelion, 20940 years.


The explanation of each constituent can be rather complex (some examples in this other answer). The different amplitudes in a day and the spring-neap cycle are related to the combination of the main lunar and solar effects. 

The differences in high/low tide from month to month are related to the next two main frequencies of oscillation. Mainly, the variations in Earth-Sun distance associated occurring in a period of tropical year. The lunar distance also contributes to these differences, but its cycle is much longer (~9 years). Also, the spring-neap cycle (with a frequency of half a lunar month ~13.5 days) will occur at different times of the month and can lead to differences in tidal amplitude if you compare the tides measured the same day of consecutive months.

(Source www.niwa.co.nz)