Do AVR can do anything ARM can't?
IMPORTANT POINT: When comparing MCUs, compare a specific model to another specific model, not an MCU family to another MCU family. Make your choosing similar, For example you can compare an ARM Cortex M4 MCU to an AVR32 MCU since they are both 32-bit-architecture based. Then you can choose a specific model from each one to compare.

Another point, You are comparing a family of MCU to a vendor-neutral architecture. Choose a vendor that manufactures ARM-Based MCUs to compare it with AVR, because ARM is an architecture, each vendor takes the architecture and adds its own features and system.

After that you can decide between ARM and AVR.

And as an answer to your question, I think yes, It worth learning. AVR is one of the popular architecture. And It won't harm to learn AVR and ARM :)