Summiting on or soon after a full moon is very beautiful and helps illuminate the landscape on summit day on Kilimanjaro without using headlamps. Nighttime photos on Kilimanjaro and on safari are beautiful with the moon in the sky.
However, it is also very bright for sleeping and stars are not as visible. If you want to get a time-lapse photo of the stars, a new moon is a better option.
To summit on a full moon, you will want to start a 6-day trek 4 days before the full moon (3 days before if doing a 5-day trek, 5 days before a 7-day trek, 6 days for an 8-day trek).