Until I started digging I didn't realize that the Arduino processors are mostly eight bit and the ability to handle numbers larger than 255 is done through the compiler! This is damn annoying as it means that Revolution Education (the people who run PICAXE) could have introduced thirty two bit variables years ago just by modifying the IDE and the compiler. OK, it would run slower, but it would still be useable.
Hmm, slower, but also use more RAM and Flash, unless used sparingly. The top end PICAXE has 16kB Flash and 1280 bytes RAM, compared to Arduino Uno which is 32kB Flash and 2kB RAM, and then there is the Arduino Mega which has 256kB Flash and 8kB RAM.
Writing compilers is not entirely trivial, Arduino leverages the gcc suite which already had AVR support in it, and the Java IDE was also not written by them, so they started with a good base of powerful software, and they have spent their effort tailoring the system for Arduino. I think that the PIXAXE software is proprietary and all written in house, as far as I know.
Unless you are doing some heavy number crunching, native 32 bit is not that big a deal, but there is Arduino Due which is 32 bit Cortex M3 at 84MHz. I think you will be pleasantly surprised by the power and ease of the Arduino ecosystem if you are coming from a PICAXE background.