Khajiit are an offensive style and Argonian is defensive. Khajiit have claws to do unarmed damage, and use one-handed combat and sneak (daggers and bows). They also have the highest sneak bonus of any race. Argonians have sneak to use with daggers as well, but don't get the extra bonus with bows. Argonians also have illusion magic for things like fury which makes your enemies fight eachother while you stay hidden or sneak up behind them with a dagger. Khajiit can use NightEye any time they want (its not permanent like Oblivion, thank god), and Argonians can breathe under water (though there seems to be a lot less under water in this game, and there's a glitch that allows annyone to breathe under water, seemingly. They both have bonuses to pickpocket and lockpicking, though Argonians have the highest lockpicking bonus of any race. Argonians have restoration magic to heal and prevent damage, Khajiit have alchemy to heal, poison, and create buffs.
So incase I lost anyone, here is the breakdown:
Khajiit: +10 Sneak, +5 one-handed, archery, lockpicking, pickpocket, alchemy.
Argonian: +10 Lockpicking, +5 Light Armor, Sneak, Pickpocket, Restoration, Alteration.
So it really comes down to how you want to play. If you want to use bows from afar to do damage and confuse your enemies (while poisoned with alchemy), then sneak up and assassinate them, or sneak attack with a sword, axe, or mace, choose Khajiit.
If you prefer using illusion magic while hidden to confuse your enemies, then assassinate with a dagger (x15 damage with sneak, who needs anything else?), use magic to heal, and steal everything in sight with improved carry weight (from light armor) and a bonus to Lockpick, go Argonian.
I personally used a Khajiit, but I play more Argonian and I wonder if I should have chosen that instead. If anyone plays as Argonian would you post feedback?
And has anyone else noticed that the word Khajiit looks like a cats paw with the dots? Write it down on paper if you don't see what I mean.