That's not a particularly easy question to answer in general terms, especially as there are a whole range of BS2 variants.
For a specific purpose, with known requirements, it would probably be possible to identify a particular PICAXE which is best suited. An answer would depend on how many I/O lines are required ( digital and/or analogue ), how many need to be bi-directional, how much program memory plus data storage EEPROM is required, and desired speed of operation. Supported serial baud rates also vary across the range.
There's no simple matrix which PICAXE variants map to but I have a comparison of PICAXE's at ...
http://www.hippy.freeserve.co.uk/picaxes.htm
Unfortunately it's a little out of date, and some commands since added to PICAXE Firmware ( SERTXD / INFRAIN2 / READTEMP12 ) isn't shown for some devices. E&OE.
The PICAXE is programmed using a BS1 language rather than a BS2 language.