What is the difference between PIC and PICAXE?
The various Picaxe chips are actually Pic chips with a small piece of code burnt in. This code acts as a bootloader for the chip, which simplifies things when it comes time to upload your code to the chip. You are restricted to using Picaxe's version of BASIC, or a flowchart based system.

Pics on the other hand are a number of different families of chips. The ones the Picaxes are based on are 8 bit chips, although there are dozens more to choose from in that range too. There are also 16 bit and 32 bit chips too. They can be programmed in Assembler or C (and various other languages if you can find compilers or bootloaders for them) using various tools available direct from Microchip their manufacturer.

In terms of programming tools there are free IDEs for Picaxe for both Windows and Linux available from the Picaxe website. To get the code onto the chip you need either a Serial cable and a simple circuit to connect to the chip, or a USB cable from the Picaxe site which connects to the same circuit. There are quite a few extra items you can get, which can be interfaced to Picaxes all from their site.

As previously stated there is a free IDE for Windows which can be downloaded from the Microchip website and used to write either C programs or Assembler programs at least for the 8 bit chips (I've no idea whether it'll target the 16 &amp; 32 bit chips I'll check later). There are various solutions available for Linux including Piklab and Pikdev. To get your code onto your chip you'll need either one of the traditional Serial or Parallel port programmers of which there seem to be hundreds on the internet, or a USB solution such as the Pickit 2 or 3 which as available from Microchip.

Although the Picaxes seem better suited if you are working with school children or have very little programming experience I'd say the Pics offer a greater flexibility and variety of products. 

You may also want to investigate whether ATMEL chips or the Arduino which is based on them offer a better fit for your requirements. I found the Wikipedia pages detailing the various different families of microchip to be very useful for getting a handle on what was what with the various types of chip. I'll try and add some useful links when I'm back at my computer.