It depends on what software you want to use, what software you want to write, how much power you have available, how many production units you'll want, etc.
If you want to do a one-off and have AC power, use the Pi. If you want to make a billion of these and fit them in a tiny box with 4 AA batteries, the AVR is going to be a better choice. But Linux + Python is easier to program than AVR firmware, so it's going to take a lot of units before the AVR pays for its difficult software development with its reduced parts cost. (Really, you should be using ARM, not AVR. It costs the same but it's a modern architecture.)
code for Linux and code for AVR are two totally different beasts. With AVR, I feel that you could just drum up a quick hack and have code running within a couple of hours. With Linux, I always felt that you need at least a couple of days to get things to going (eg, having a root fs structured, choose a kernel, make sure that your memory map is set up correctly etc). Maybe, I have been away from embedded Linux for too long and the goal post have shifted towards a better location.
If you want to do a one-off and have AC power, use the Pi. If you want to make a billion of these and fit them in a tiny box with 4 AA batteries, the AVR is going to be a better choice. But Linux + Python is easier to program than AVR firmware, so it's going to take a lot of units before the AVR pays for its difficult software development with its reduced parts cost. (Really, you should be using ARM, not AVR. It costs the same but it's a modern architecture.)