Just my two cents.
From a player perspective I see them as two very different things. Half-dragons are a template, they're much stronger than the average person (level adjustment, depending on how that is done). They're more draconic and by that I mean they have tails, wings (in pathfinder, or if large in 3.5), stronger scales, (already said) stronger, breath weapon. Dragonborn (or dragonfolk as I call them) are dragon-people. They lack the outward appearance (tails, wings) and can't do a breath weapon. Honestly this is part of what bugs me about dragonborn and why I'd prefer to use Lizardfolk.
Now, from an in-world perspective (as a character) then I can understand having a harder time telling the two apart. But then again that doesn't mean very much to me. I think that most characters would have a hard time distinguishing a ghost and a spectre, for example. But even then the visible qualities (tail, wings) could show through. Also in how they carry themselves - dragonborn (per 4e as I understand it) are honour-bound. Half-dragons are just that, half-draconic and so they don't act a certain way universally. Dragonborn have a RACE and usually that means that you'll see more than one of that race at a given time. That means a half-dragon can be identified by being solo, even before he spews fire and takes off into flight. Beyond these points I guess I could see a half-dragon dressing up and disguising himself as a dragonborn, I have no issues with that. I can still see the difference from both in character and out of character perspective.