I think there is more than one reason.

(US Only) There is a lot of obvious anti-war bias for suspect reasons, i.e. France and Russia, and the local anti-war folk don't want to be perceived as in their camp or subject to an 'I told you so' in French.

A lot of people are in favor of the goals of the war, but making anti-war noises as the media try to convince us the 'coalition' is bogged down. So with every success tied to the Iraq war, this sort of protest is damped down in a hurry. As an example, without the Abu Ghraib scandal, there would be much less anti-war noise than there is now.

And to my mind the main reason (in the US).
Way different from the Vietnam situation THERE IS NO DRAFT. The huge protests against the Vietnam war were driven by youth who didn't want to serve, and their girlfriends and friends and parents who didn't want to lose them. Now we are dealing with a volunteer army. The core of this kind of resistance would come from the family of National Guards people who are facing extended service, and so far there is negligible resistance among them.

Without citing sources, I have heard that in fact, with all the Vietnam demonstrations that went on in the 60's and 70's, a majority of Americans did in fact support the Vietnamese 'police action.'