The main difference is the programming environment.
With Picaxe controller you use a programming editor which support a Basic-like language and the hardware has a built-in programming interface (which is the reason they cost that much). With PIC micro controller you have to use PIC micro Assembly Language programming or you can also get a C language compiler. Assembly language is much harder to use and it needs a lot of knowledge of the hardware of the specific PIC micro controller you are using.
In the hardware side, they are the same (actually Picaxes are PIC micros) but Picaxe micros come "ready to use" while you have to set up the PIC micros, which again requires hardware knowledge.
In others words, if you want to use PIC micros you have to read and understand the technical manual of the specific micro and to learn to program in Assembly or C. If you have some experience, this is not necessarily a big deal, but if you are a beginner, this could be daunting.
You can download the technical manual of all PIC micros from:
http://www.microchip.com
If you think they do not look too scary...go for it!
You can lower the cost of your projects a lot with some invesment in time and study. In any event you will need a new programmer and a some kind of carrier board which are about $50.
If you need additional support I recommend you to get the book "Designing Embedded Systems with PIC Microcontrollers" by Tim Wilmshurst. There are tons of garbage books about PIC micros, stay away from them, you will not learn anything from them.
Another alternative is to get a "external" Basic-like programming interface, but, even when it is an one-time buy, it is very expensive, $250 for the compiler plus $80 the programmer.
http://www.melabs.com
May be at this point you think that Picaxe micros are not "that" expensive...:-)