So what is the difference between ARMmbed and an Arduino?
Arduino is based on 8bit Atmel micro controller.
Arduino is a complete development system, with complier, downloader and open source board designs.
Arduino emphasis is on user friendly, especially to get up and running quickly.
Aimed soley at hobbiests
Embedded ARM is based on Cortex M series of processors.
( Cotrex A series are the processors used in mobile phones, tablets and sort of the Raspberry Pi)
There are at least 5 development platforms for the Cortex M seires (editor, compiler, downloader and debugger).
There are something like 4 main chip vendors (NXP, TI, ST, Freescale - have been bought by NXP so do not know how much longer fs devices will be around) that make M seriex processors (note I say main because there are many more that produce Cortex M devices but are lower volume or niche products, note that Atmel has joined the fray and now produce M series microcontrollers)
People working on Cortex M are expected to know what they are doing (there is not so friendly initial code framework unlike the Arduino) and develop the boot up infrastructure (Arduino this ia all done and no need to worry about it)
Much wider variety of development boards with different levels of device integratio and debug support.
No concept of 'shield' adapters for most development boards, users are expected to be expert electronics developers and make their own adapter boards.
Aimed mainly at commercial market for product development, though there is some effort to capture the hobby market particularly by NXP and ST
I could go on