Conner's 'costume'- jeans and a T-shirt- seems odd at first, compared to pretty much every othersuperhero. But considering Conner's identity issues, it makes perfect sense. He's trying his hardest to not be Superman, in any way possible. He wants to be his own person, and his choice of costume reflects that.
Further than that, Conner's real identity is Superboy. He's wearing normal clothes because being Superboy is when he takes his costume off.
