In your opinion what are the advantages of periodized exercise programming?

It is a tremendous way to help people to train just below the overtraining point. It also adds stability to a program. There are many ways to periodize programs, however. In the literature though it doesn't appear that undulating or linear has a significant advantage over the other. I think there's place for both in a year in my training.

I think periodized training keeps clients engaged and progresses their training while reducing their risk of injury. As a trainer you're able to see their progress, increase intensity gradually and work on the areas that tight/weak in a progressive manner, while still helping the client achieve their goals. As a client, they don't get bored with their program. They learn a new exercise and then how that exercise can be changed to make it more challenging for them as they get stronger, more flexible, etc. Its a win/win for everyone!

I believe first and foremost that periodization can help just about every one of your clients regardless of individual goal as far as reducing injury and improving fitness level. It's a powerful training tool which will help achieve greater results over time

planned progression

gives a bigger picture then breaks it down into managable segments
such as and Olympic training cycle (4years)
an annual training cycle
mesocycles
microcycles
then allows for manipulation of acute variables within each cycle according to goals

no or minimal plateau (if used properly)

reduced risk of overtraining (if used properly)

minimize repetitive overuse syndrome (if used properly)

minimizes the cumulative trauma cycle (if used properly)

allows for proper recovery of connective tissue (if used properly)