I think as long as you maintain over a 3.5 GPA at UCSD your eligible for the honors program (I'm not sure what the GPA requirement is to be admitted straight from highschool). So no, it's not really THAT big of a deal (Highest Honors is however, but it requires a senior thesis if your willing to write one), that said, if your looking to go to a top med school you will obviously be favored over someone with the same stats but no honors and perhaps someone with even slightly better stats. I don't know anything about NYU, but I'd recommend asking Astrina, shes a frequent poster on here and went to UCSD and then transfered to NYU med school, at least I think thats what she did....
EDIT: "- currently: in new york, at columbia pursuing a PhD in biochemistry" Nevermind.. she goes to Columbia for biochemistry