CVT :
No shift-shock (You won't feel the car shifting gears, rides like a scooty)
More fuel-efficient
Very sensitive to abuse - you should maintain it with utmost care. Only the manufacturer-specified CVT oil should be used and transmission oil should be replaced in a timely manner.
Won't last as long as a regular auto which is maintained with similar care would.
Some Toyota CVTs had to be replaced at 100k-150k mark even though they were maintained properly (Source - Nilwala)
Conventional auto
Not as much comfortable as CVT - You can feel the car changing gears
Not as much as fuel efficient as CVT
Durable - most Japanese autoboxes would surpass 300k km mark without major repairs as long as you replace transmission oil every 40k km
You can use any transmission oil which meets manufacturer's specifications
Can be repaired locally unlike CVTs