Tower Controller described a confused clearance event that resulted in an unexpected route turn after a hand off to a subsequent facility.

Narrative

While working Ground Control; I had two flight strips on the C550 separated by 45 minutes; both to FRG. The two clearances were the same except for the departure time and squawk code. The previous Ground Controller had read the C550 his clearance. Later; the Pilot called up asking for his new clearance. What I didn't catch was that he said new clearance to BWI. This was partially because of where he was parked on the field; his radio was very scratchy; and partly because I had the other clearance sitting in front of me. I assumed that he wanted the newer clearance to FRG; and told him that the clearance was unchanged except for the squawk code. In his read back; he said cleared to BWI; route unchanged; and I told him read back correct; again not catching him saying BWI. I had no flight plan to BWI on him at this time. He taxied out and departed with no problems. After he departed; a flight plan printed out for him with BWI as his destination; and I figured that Approach had changed his destination and still left us as the point of origin; as they sometimes do. A few hours later; someone from a Center out East called and asked us where we had cleared the C550 to; because the Pilot thought he had been cleared to BWI; but he was squawking the code I had given him to FRG. It was at this point we went and listened to the tapes; and learned that I missed him saying BWI. Apparently the Pilot had started to fly a different course than what the Center was expecting and they were trying to find out why; and the Pilot said he was flying the routing of the clearance to BWI. Recommendation; I first should have had the aircraft pull out from the building he was behind so I could hear him better. I was using a handset due to low traffic volume; and could possibly have heard him better if I had been wearing a headset. I should have called Approach to ask about the strip that printed out for the C550 to BWI to verify that they had changed him; because the mistake could have been caught at this point. I should have verified what airport FRG was because I assumed that the Pilot was asking for FRG when he called me for the new clearance; since those were the strips I had in front of me. I should have questioned why the Pilot wanted the new clearance when I thought they were identical except for the squawk code.