I disagree with having a "poll" on this. I would retain both functions were I the site owner .. which I'm not. Like Dave I am in favour of both and that they are not anonymous. I use them to indicate agreement or disagreement when I have nothing to add, or my view has been already made clear or is most probably known. I see no point in posting "I agree" if I have nothing to add, nor repeating my reasons for disagreeing. As I said on the other thread, I use them like I would nod or shake my head in a busy debate.
And why should a perfectly legitimate option be taken away because some people may not use it sensibly, appropriately, but in an indiscriminate way or to score points/play games ... according to others?
We might as well say people may not post either if they can't be trusted to do so with civility and respect, and without losing their tempers or sniping at each other, or posting comments to flame a response ..
Oh wait .. we have a means of dealing with that 
We have use of the report button and the moderators deal with it. Seems to me that if overuse or style of use of the like/dislike buttons offends anyone, it is the moderators who should be approached, privately
Caryl