There are some misconceptions out there about weather and moon phase when it comes to the elk rut.
The simple truth is neither affect the timing of the rut. The amount of daylight (called photoperiod) is what triggers the rut. However both will affect how the elk behave on a given day. If the weather is hot then the elk are active at night when it's cooler and tend to be less active all around. They are more active when the weather is cooler, but the rut timing hasn't changed.
If the moon is full elk tend to be more active at night and are wore out by the time the sun rises. In short the moon can be a bad thing.....pray for clouds. Having said that you can still be successful elk hunting during the full moon phase because you can go after the elk unlike whitetails where you're usually waiting for them. They will still repsond to calling if you get close enough to the herd in the timber.