They're very similar on the playstyle but have major differences too. Akali is one of the best chaser if not the best in the whole game, she has mostly single target spells, she can cloak, has natural spellvamp and can melt any squishy very easily. Akali have one of the weakest early game, but once you hit 6, you are unstoppable. Katarina is more an aoe champs, she can deal massive damage to the whole team, she has one of the most aggressive early game, a nice escape mechanism, cooldown refresh on kills. They are both vulnerable to cc but a big part of Katarina strenght resides in landing a good ult, if she gets shutdown while casting it, she will most probably fail. They are both very solid champions above average and good pentakillers since kills reenergize them (akali gets ult stacks and katarina gets cooldown lowered). If you want a chaser that focuses important targets go for Akali, if you wanna jump in the fight and deals massive damage to everyone go for Katarina. They both use no mana, snowball like crazy but also become weak quickly if they don't get some early kills. No matter which one you pick, you can't really make a bad choice.