Designing a Audio Spectrum Analyzer
Well, most people are good at analog circuits, or MCU/DSP stuff.  Not many people are good enough at both.  So...  Do what you know better.

The DSP approach will be more flexible, and would allow you to do cool things like: change the decay time, do both a peak and RMS reading at the same time, etc.  The response time will be slower than an analog approach, but still faster than what your eye could perceive.  

As Ben Jackson pointed out, the LEDs will draw more current than the rest of the circuit.  Assuming that you're driving each LED with 20 mA, and you have 31 bands with one LED on for each band then you're pulling 620 mA just for the LED's.  

As for which approach would be less expensive, that all depends.  If this were a commercial product then the DSP based approach is by far the cheaper one.  But the economics of working in your basement changes the whole equation.  For example, if you need to spend $100-500 on DSP/MCU development tools then that's an issue.  So in the end, we can't say which is cheaper-- only you can.

So this brings me back to my original point:  do whatever you do better.  Do whatever one makes sense for you-- and just know that might not be what makes sense to others.