That's weird. I've had the opposite experience. I love both champions and play them in ranked, but I've never lost to Kat as Akali.
If we take out personal skill as a factor, I think anyone can objectively see that Katarina triumphs in team fights at the expense of her laning phase and one on one capabilities - something Akali easily takes advantage of post-6. Akali's sustain and one-on-one power far surpasses anything Katarina can trade.
1. Katarina generally needs to poke the enemy low enough (bout 50 percent) before a full on commitment to get a kill. Akali quickly recoups any lost health by mitigation (shroud) or spell and health vamp (built in + supplemented with items).
2. Katarina needs a stationary target to fully benefit from her ult. Akali is slippery. She can ult out of, or just walk, Katarina's ult by targeting a minion and ulting back in with 1 charge to spare.
3. Katarina needs items to do a lot of damage with her ult. In the levels of 6-11, Katarina is still ramping up. Her power comes from her capacity to farm easily and acquire the funds she needs for bigger items that provide AP or Magic Pen. Not counting kills in other lanes by roaming or jungler help, Katarina just doesn't have the juice to take out Akali. In contrast, Akali's power AND mobility take a sharp upturn at 6.
In short, it's my opinion that notwithstanding player skill, roaming, and jungler help, Katarina is designed by Riot to ramp up over the games time limit and become strong for team fights.
In contrast, Akali, a champion gated by energy instead of cooldowns, is designed by Riot to deliver upfront early and mid-game damage at the expense of her late game. Late game is when anyone is liable to be running around with pink wards. Also, champion health and resistances are increased much too high for Akali to one shot and isolate - her preferred method of killing. Hence, her lane dominance.