If you purchase 'blank' PIC chips they will not work in the PICAXE system, as they do not contain the PICAXE firmware. Therefore always buy pre-programmed 'PICAXE chips'.
So the PICAXE also needs a bootloader before it accepts sketches, via the serial port. In both cases you can use the ICSP to upload things (including the bootloader). In both cases, once you have the bootloader you can upload further sketches via the serial port.
PICAXE is quite a bit different from Arduino.
PICAXE really provides a language and uses an interpreter for tokenized code. You cannot extend the language.
You can only do the commands provided in the language.
Arduino is not really a language but a framework. Since it uses C/C++ it is infinitely extensible.
Where you can run into issues on PICAXE is that it you cannot extend the language so if you want
to support some new device that is not supported by the language, you have implement it by using
the interpreted language constructs which will not be nearly as fast as the machine code you can get
when using C/C++
On the positive side, PICAXE includes some excellent debugging tools - which is something that the Arduino does not have.