Currently each card I have has the activity relating to the card on it. As I have the stream on the left I don't need this. Is there any way to disable this without also hiding comments?
There is not currently a way to do this, no.
You can't do this in trello but you can hide activity using a Greasemonkey / Tampermonkey script from here - http://pastebin.com/eSxriXg6