The major benefit of taking a 9-day route, or adding additional acclimatization days to extend your route to 9 days, is to acclimatize your body to the high altitude, thus improve your summit success chance. One of the biggest hazards affecting Kilimanjaro climbers is altitude sickness, but walking slowly and spending more days climbing gives your body time to get used to the altitude.
The flipside of this is that each day spent on Kilimanjaro can be stressful for some climbers, due to insomnia, feeling unwell or the stress of camping. This may weaken a climber and outweigh the benefits of improved acclimatization. It is not yet clear whether 9 days significantly boosts your success chance compared to 8 days. Both 8-day and 9-day routes have a summit success chance of close to 100%.