I don't know anything about battles, but I'm in the Jeannie camp. I think their powers are pretty much the same, meaning TV-magic which can do anything. Jeannie teleported into space, traveled back in time, turned people into all manner of animal and object, but so did Samantha. Now, the one weakness that Jeannie has that Sam doesn't is the bottle. Whoever owns the bottle, can command the genie within. Of course, classic nature of the d'jinn is one of manipulation and deceit, so most of the time, the wishes they granted ended in tragedy. Jeannie was a romanticized update on the classic and she was only vengeful from time to time. Come to think of it, she had a mean streak to her that Samantha certainly was missing. I recall many an episode where she'd wisk Roger or even Tony into some burning death device with just a blink.
Next up: "Jeannie", Jeannie's evil sister versus "Serena", Samantha's evil cousin. (aka "All Brunettes are Evil")
Jeannie's sister is also named Jeannie - Samantha's cousin is name Serena (Usagi?!)
On a side note, the original theme song to I Dream of Jeannie is still among my most favorites. It's different than the one most people know of since it only ran during the first season which was in black & white.