To some extent, smaller rockets require better weather conditions in order to launch. Though, it depends on the instrumentation on the rocket more than the size.
When Apollo 12 launched during a thunderstorm, the Saturn V was hit with several lightning bolts. The Saturn V was built with the ability to launch in virtually every weather condition except the end of the world. It was not necessarily the size of the rocket that mattered, but the resilience. Being a manned rocket, there were basically backups for backups, making the Saturn V extremely resilient to outside forces.
A wind shear is a rocket's worst nightmare. The rocket engineers know the limitations of the rocket and may give a GO for a launch, then during launch, the winds violently change, knocking the rocket off-course.
Agencies prefer calm conditions because it does not strain the rocket's guidance. Given the choice between barely using the stability assist or heavily using the stability assist, the launch specialists will definitely go with barely using the stability assist because if the SAS (stability assist system) were to go haywire, the next blow will send it downwards.
The capability of the SAS is based on the purpose of the rocket. For instance, the Falcon 9 is meant to go to LEO (low earth orbit.) Conditions for low earth orbit reset almost every ninety minutes, giving SpaceX a more flexible launch window. The Saturn V was committing to a moonshot. In order for the Saturn V to launch, NASA had to take the moon phases and the moon's orbit into account. The moon's orbit (a day) and the moon's phase cycle (twenty-seven days) overlapped provide little opportunities to land on the moon, requiring a less flexible launch window. In order for NASA to deal with the constricted launch window, they had to make a rocket that can deal with any instances of harsh weather. Hence, they packed the Saturn V full of instrumentation.
P.S. You can cross-check moon-phase cycles with Apollo launches using this moon phase checker and a list of Apollo launches. You'll find that most launches took place either close to the full moon or first quarter or smack dab in the middle of a full moon.
- Saturn V Instrumentation Unit
- "Challenger encountered the first of several high-altitude wind shear conditions, which lasted until about 64 seconds...The wind shear caused the steering system to be more active than on any previous flight." NASA JSC