I have been playing this game since season two. My mains have always been Akali and Katarina. Mostly Akali until recently because I have finally gotten bored of her.
Akali has more burst, once she has gunblade she destroys people, and keeps healing up. Has three dashes and lots of mobility and goes invisible. You cannot quite count on the invisibility because of pink wards, but stealth is getting reworked soon so it may change. Most of her burst comes from Q, she applies a mark and upon auto attacking the marked target, she deals a lot of damage.
Katarina has one blink, and much longer cooldowns, but upon getting a kill or assist all of her abilities reset by 15 seconds. She has more ways of utilizing her combos, and has much higher outplay potential. Less burst because of the resetting mechanic. She is likely to be one of the bigger reworks coming out in a couple of months. She seems easy on paper, but she gets harder as you rank up.
I personally prefer Katarina, and is now my favorite champion. The difficulty of the champion scaling depending by how good I am is fantastic and always keeps her fresh. I don't believe I'll ever get bored of her.