The PICAXE and PIC families of microcontrollers can easily be confused as one is based upon the other.  Even though they are related there are significant differences that must be considered when deciding which to use.

A PIC processor is manufactured by Microchip Corporation in Chandler, Arizona.  There are dozens of different PIC chips.  They can be programmed in a number of different programming languages including assembler, C and BASIC.  A special programmer is needed to put the program onto the PIC.

The PICAXE is a PIC chip that has had a special program installed on it that allows it to be programmed using a very simple serial connection, not an expensive programmer.  The PICAXE can only be programmed in BASIC.  The software for the PICAXE was developed in Great Britain and was designed for use by school students.

Programming the PICAXE in BASIC
The software that is needed to program the PICAXE in BASIC is a free download from PICAXE.com.  The language has been revised and updated many times over the years and has become very capable and mature.  It includes a simulation mode that allows you to test a program without even having a PICAXE chip.  It also includes a terminal connection that allows you to have the chip send values and other information to the computer as the program is running. 

The manuals for the PICAXE programming language are free downloads as well.  There is also a very helpful on-line community that provides help and support.

The programming editor runs on Windows computers.  There is a simpler program that can be used on a MAC or Linux computer.

Programming the PIC Processor in BASIC
A special program must be used to program the PIC family of processors in BASIC.  The most popular version of BASIC is called PIC BASIC Pro.  It costs about $250 per copy.  It is a very capable programming language that allows you to produce very sophisticated and complex programs.  The program only works on a Windows computer.  I am not familiar with a similar program for MAC or Linux.

Programming Hardware for the PICAXE
The PICAXE was originally designed to be programmed with nothing more than a connection to three pins on a standard computer serial (RS-232) port.  If you still have an old PC with such a port you need nothing more than a DB-9 connector to plug into the serial port.

Since most modern computers no longer have a serial port you need to purchase a USB to Serial adapter.  A number of them are available for less than $20.00.  I have described a number of such adapters here:
http://www.trainelectronics.com/PicaxeSpeedController/SparkFunUSBCable/index.htm

Programming Hardware for PIC
You must use an external programmer to program a PIC.  There are a number of options that vary in price from $30 -  $100.  I have used programmers from MPLab including the PICKIT and their USB programmer.

Recommendations -- PICAXE or PIC?
If you are just starting out with programming microcontrollers I would recommend choosing the PICAXE.  The start-up cost is much less and the software is much easier to use.  The on-line support community is always there to guide and encourage new users.

The PIC processor is a better choice if you need to write long and more complex programs.  In most instances the PIC can hold two or more times the programming code than the PICAXE can.  The programs also can run faster and can react to inputs more rapidly.  On the down side the programming language is much harder to learn and you must set a number of configuration parameters to set up programming for each chip. 

There is an extensive on-line community but they tend to be less willing to guide beginners through their initial learning. 

For more information about the PIC processor and PIC Basic Pro see: http://melabs.com/
For a list of programming commands see: http://melabs.com/resources/pbpmanual/

PICAXE information is available here:  http://www.picaxe.com/