I’ll do my best to make this straightforward. Somebody correct me if I’m wrong, as I’ve had limited experience with these hobby systems.
Both Arduino and Basic Stamp are just regular microcontrollers with a boot loader on them that allow you to program them through a regular serial connection.
The Basic Stamp runs an interpreter on the chip that takes the code you’ve written in the Basic Stamp development and environment and translates it as it runs. This is very slow, but allows the development environment to be very easy and controlled.
The Arduino development environment is much closer to a standard environment for microcontroller development. Code is compiled into a program that the controller runs directly. The Arduino has become very popular as it is fairly powerful on it’s own and can work as a standard microcontroller stepping stone. The BASIC Stamp can be useful, but it’s essentially a dead end. The language is not transferable to another controller and you can’t really take anything away from it other than the most basic microcontroller concepts if you later want to move to something more powerful.