With Zooms, focus breathing tends to be worse than with many primes. This means you may see a zooming effect as you focus. Its probably not a huge issue, but at close distances, it is noticable.
Secondly, consumer zooms are not parfocal, which means the focus will change as you zoom. You do not notice this when taking stills, because the autofocus takes care of it. However, with no autofocus, it will be apparent. As long as you dont zoom in and out a large amount while taking a video, it might not be noticible.
Thirdly, using a zoom frequently in a video makes it look amateurish, but, thats only my opinion.