As I see it the advantages to renting are-
You can change skis for current conditions whenever you need to.
You can change skis for the terrain available.
No self maintenance (some people enjoy this tho).
No additional maintenance costs (unless you abuse the rentals).
No additional baggage cost on planes.
More space in your car.
You can try out diff length/width skis to find out what you like and narrow it down a bit when you do come to buy. (helpful if you can't demo).
As you progress you can rent better skis without huge investment.
The skis may have been skied out and lost their pop.
They might be in otherwise poor condition.
Queueing in the shop when you could be out already.
Not getting the continuity of using the same skis all the time and learning what they/you can do.
When you add up what you've spent you might have been able to buy some in the sale or slightly used. (flexible seasonal rents seem to be the most cost effective route)
I rented skis for four years while I had lots of lessons figuring I'd buy some once the learning curve slowed down. This was the right decision for me. You could be different as your getting at least double the time on the mountain in a season.
When renting I used to avoid the shop when it was busy so I could pick the brains of the staff (we used to ski diff areas everytime so never knew what we were likely to need).
Have fun, sounds like you have well and truly been bitten by the bug!