Recording at the same time doesn't matter...
If I can't follow your "rendition" and it throws me off, then it will inevitably throw others off.
There's also certain elements to the way the original drums were recorded, for instance; the build up on the toms, building up every time before the chorus will be dull and won't give the same effect as being silent and then having it coming in, loud and proud.
Another element is the 2 quaver beat during the verse which in the music video, Hayley claps along with.
I've spent 6 years in music, half of it was spent being a drummer and I've spent all those 6 years listening to other peoples "renditions"... It's all well and good claiming to be "original" but if it's not suited towards the style, others will think the same.