It's the same kind of difference. It's just that Sasuke in anime was holding his hand and chidori upwards and Kakashi is holding them downwards. Anyways in anime you can clearly see Kakashi is holding under his hand something of a "ball" shaped raiton. Also the top of his hand isn't covered with raiton, clearly the raiton is concentrated only under his hand.
In the manga you can see that was never the case. Kakashi indeed also stretched his fingers so it looks like he's holding something, but in reality he isn't and raiton is formlessly covering his whole hand, instead of being concentrated into the ball which you hold.