For simple tasks where speed is not an issue, the picaxe will win hands down, as its simple to program, comes in a range of chips from 8 pin (14,18,20,28) to 40 pin and the built in interpreter has many functions you just call in program without the need to understand lines and lines of tricky coding.
Excellent chip to get your feet wet in the world of using micro controllers.
Arduino is a faster chip for more serious and compicated projects, where speed and memory space are needed more, but the cost is greater and the programming language is harder if you have no background in programming.
My advice would be to start with the picaxe and when you out grow their ability, and they can no longer fill your needs of programming ability than move up to the Arduino.
If you have never used micros yet, than it will take you a few years to out grow the picaxe, as they are getting faster all the time with larger memory space, for example the 20x2 chip can be run up to 64 meg which is well fast enough for most applications.
Pete.