An A319 Crew flew the incorrect aircraft on their first flight of the day.

Narrative

We mistakenly flew the wrong aircraft on our first flight. We reported to the gate at XA:20; the release was still being printed; so we boarded the aircraft and started with cockpit setup. The aircraft had already been de-iced and fueled properly and the associated paper work for both listed our flight and aircraft. The security sheet was on center pedestal marked as our flight. All of the local paperwork matched the aircraft tail number. The maintenance log was reviewed and checked. The flight release arrived and the First Officer began programming the flight plan into the MCDU. I reviewed the flight release and missed the difference in aircraft tail numbers AB(X) versus AB(X-4). After push back we contacted Local Operations for our final weight and balance; and we were told it had been sent to our aircraft; but they would ask Load Planning to resend it. We also sent a message to Dispatch through ACARS about the need for the ACARS weight and balance. We never received a response back from Dispatch. So; with the brakes set; I called Dispatch on my cell phone about the problem and agreed to have Maintenance meet the aircraft at our scheduled destination. We radioed Operations for verbal numbers; programmed the MCDU; and departed. Our flight was uneventful. We threw away all of our paperwork from the first flight and proceeded to the crew lounge for our 2:15 ground time before departing. While waiting for our next flight; I received a phone call from the flight Dispatcher; telling me that we took the wrong aircraft. What can we do to prevent this from happing in our system again? For myself and the First Officer; we will redouble our detailed review of the paper work; and I'll have the First Officer read back the aircraft number when he reads me the V-speeds and Flex numbers from the ACARS weight and balance. While I'm sure it will never happen to me again; one recommendation is to add an additional item to the 'Safety & Power On' checklist.