Here is my comparison of Ahmed Baba and Annie Oakley. I don't remember much details of Ahmed Baba, so this is at the top of my head.

MOAT:
Ahmed Baba had the "Rope Swinging" moat and despite having an easy objective, it was kind of long. Annie Oakley had the Babylonian wheels moat where the players had crawl through the tubes to make it across. This was both a more unique and more challenging moat.
My vote goes to ANNIE OAKLEY.
LEGEND:
Annie Oakley's legend was easily better than Ahmed Baba's legend. I love Wild West stories, and Annie Oakley's legend was much more interesting. It talks about how Oakley was sharp with a gun, and how she shot a cigarette out of a Prince's mouth. I mean how is that not badass? All Ahmed Baba legend is about is how Ahmed Baba is captured and how he tells the caliph that he must be lonely standing behind a curtain. Something like that.

My vote goes to ANNIE OAKLEY.
ARTIFACT:
I want to do a separate section for this since I think it is a little bit important. I like the Very Tall Turban's name and artifact design. It was pretty cool, even if the Turban wasn't tall. The Leopard-Skin Cloak was kind of lame and had nothing to do with the legend. And it looked way too similar to the Imperial Purple Robe.
My vote goes to AHMED BABA.
STEPS OF KNOWLEDGE:
Both Steps were close, but I recall Annie Oakley's SoK was even closer. There was a three-way tie on the second to last step. And the questions asked in Annie Oakley were more interesting.
My vote goes to ANNIE OAKLEY.
TEMPLE GAMES:
Even though Chandra and Nick swept the temple games, I loved the temple game sequence in this episode. Each of the temple games were very relevant to the episode's legend. I liked the "horse spinning" game where Chandra didn't even fall off once and I especially liked the "Ace in the Hole" game where Nick and his opponent had to put balls into the ace with the use of a trapeze. And the temple game interviews were more memorable too in Annie Oakley. In the Ahmed Baba temple games, I was not too crazy about the competition either. I mean just watch how Jamie struggles in her temple game. I did like the rare minaret team game only used in this episode and Ivory Elephant, and the Green Monkeys had an amazing come back, but other than that these temple games didn't have much going for them.
My vote goes to ANNIE OAKLEY.
TEMPLE RUN:
I am going with the episode that ended with a loss here. Chandra and Nick were a great team in the temple despite the mistakes they made. As a frontrunner, Chandra made good time all the way to the King's Storeroom. She really should've avoided the Mine Shaft elevator, but her fate with the one room away guard wouldn't have changed. Nick did a good job of avoiding her mistake, but he had no time to do anything. Shane was just average in Ahmed Baba. He made the same mistake as Chandra did, but he got lucky that there wasn't a temple guard on the top floor of the temple. And Kirk was extremely annoying in this episode too. Shane, Shane, Shane, Shane...

Overall, I love the Annie Oakley run more because the performance was better and there was more interesting things about it like Chandra completed the Secret Password and Nick putting the key to the King's Storeroom in front.
My vote goes to ANNIE OAKLEY.
Overall, I have to say that Annie Oakley is easily the better episode. Even our episode reviews show that Annie Oakley was the best of the two. This is just my opinion, but I think Annie Oakley had more going for it.