In my experience Buddy Tape is better if you want a stronger hold for the graft and for a longer time (i'll use others materials if i need a stronger hold but i won't use anything else to cover the buds or when i do my chip-bud grafting).
Buddy Tape will not biodegrade as quickly as Parafilm does. I really stopped using Buddy Tape when i found Parafilm.
Now i can cover my buds, protecting them from lack of moisture and they can still breathe (this will not happen with Buddy Tape) and when they are ready they can break the parafilm easily, even if i forget to watch them, unlike other materials.
I have several chip-bud grafts i did in fig trees in October. The grafts fused okay in 2-3 weeks but the buds went immediately into dormancy. The Parafilm is still in place, protecting them and with the warm of spring they will start to grow freely.
When i used Buddy Tape, i would not cover the buds or, if i do, i would have to remember to cut the tape before the warm weather.
I also had more problems of fungus developing inside the Buddy Tape while they aren't a problem with Parafilm (even with lots of moisture inside)

Without a doubt the success i am having grafting figs is due mainly to the use of Parafilm, so i can't recommend this material more to who wants to grafts figs (especially when chip-bud grafting)