Difference and compatibility between AVR xxxA and xxx/xxxV series
The ATtiny13 is the original version and uses a different process technology than the ATtiny13A. The A-suffixed parts are fabricated with a low power process marketed as "picoPower", and the main difference is that they generally consume less power at the same voltage and frequency. 



The difference between the ATtiny13 and ATtiny13V is more subtle. Unlike the A version, the ATtiny13V has the exact same die as a "standard" ATtiny13. The difference instead lies in how the parts have been binned after testing: V parts can only reach 10 MHz but will work at just 1.8 V, while "standard" parts can operate at 20 MHz but only go down to 2.7 V. The more modern ATtiny13A can go to either extreme. 

  

The part of the product number that comes after the hyphen is just the semiconductor packaging being used.  

Regarding differences from the point of view of code compatibility, I see no reason why the ATtiny13A would not be code and binary compatible with programs written for the ATtiny13. However, the reverse is not strictly the case: While the instruction sets and most peripherals are identical, the ATtiny13A has the extra registers PRR (Power Reduction Register) and  BODCR (Brown-Out Detector Control Register).
You need to check your code for any accesses to those. If I were you, I would make sure that all accessed peripherals still work identically in the ATtiny13, as the two chips do after all use different silicon and nobody can really guarantee compatibility.