Wearing a brightly coloured jumpsuit with a great big target on your torso probably doesn’t seem very practical from an urban camouflage point of view, but for Steve Rogers, it’s perfect. Why is he wearing the old costume? So people will know that he’s Captain America. He’s trusting that people will see him and rally around him, because that’s the whole point of Cap.
That Cap/Falcon conversation (“How do we know which ones are the bad guys?” “They’ll be the ones shooting at us.”) isn’t just a snappy one-liner, it’s a tactical decision. In a war where no one is wearing a uniform and anyone could be the enemy, Steve is consciously choosing to represent one side instead of sneaking into SHIELD headquarters in disguise. So not only will he be able to tell who the bad guys are because they’re shooting at him, everyone else will be able to tell as well. Loyal SHIELD agents will know that if you’re shooting at the dude in the red, white and blue superhero costume, then you’re the enemy.
There’s also the possibility that the vintage Captain America costume helps Bucky regain his memory for long enough to stop murdering Steve in the face. Steve’s 21st century hair, clothes and uniform would all render him well-nigh unrecogniseable to Bucky’s distant memories of their life together, which are almost entirely subsumed by the Winter Soldier’s conditioning. Cap’s old wartime costume is the only thing that stands a chance of seeming familiar.
— Costume and design in Captain America: The Winter Soldier.