It's hard to compare ARM to AVR.
I assume you are talking about AVR8. ARM is a 32bits processor, the AVR8 is 8bits. I won't get into details (PM me if you want to know more) but basically an ARM is closer to a desktop computer. It can run a complete OS (linux and probably windows CE) and has a lot of processing power, compared to the AVR8 which is limited (memory space, processing power, ...). A LOT of embedded systems (cellphones, mp3 players, ebooks...) are based on ARM.
If you are talking about AVR32, then it's a different architecture, closer to the ARM. I think AVR is suitable for small projects and a good replacement for 8051 and 8 bit micros,but for larger projects that need RTOS,FS,TCPIP,USB,...... it is better and simpler to use ARM7 based micros.
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