IMO, you are fighting different problems. A microprocessor VS an actual computer based mini controller are about the programming side of it. A micro controller is really good at doing one thing over and over and over. A full blown computer is good at higher level networking and processing functions such as touch screen interfaces, USB based sensors and other devices.
Again, a Raspberry Pi is a computer, and yes, it has some microprocessor like IO pins, but you are writing an app that runs on top of the OS, and then can sense and manipulate IO. It takes time from power on for the OS to boot, and then your app to begin running before the machine can do any "work".
An Arduino or generic AVR microprocessor or the PIC processor family (Picaxe is just the firmware on a PIC processor) boots instantly and begins running your code nearly as soon as powered on.
Again, where the real problem lies is learning programming.
For a mini computer based setup, you might run Python or some other language as an application on top of the operating system to create your control function.
For Arduino/AVR, that's also the name of the Integration and Development Environment (IDE) where you write your program code and that in turn is linked or compiled to the machine language firmware that runs on the AVR microprocessor.
For PIC or PICAXE, you are using their respective IDE programming tools and learning that system.
Microcontrollers like PIC, PICAXE, AVR, "Arduino", they boot nearly instantly and take extremely low power to run. Mere milliamps at 5V.
A full blown mini computer like a Raspberry PI can take up to 2.5A of 5V power, take time to boot, and then your app runs on top of the operating system.