Which family of ARM processors should I learn: ARM7TDMI or Cortex M3?
You partly answered the question yourself: Cortex is a newer, more advanced core. It's more powerful than the ARM7TDMI and at the same time uses less power: about double the DMIPS/mW.
The following table gives a nice overview of the differences (from this white paper):  



(Apart from performance and power consumption) the up to 240(!) interrupts and the much improved interrupt latency make the Cortex a far better choice for real-time systems.