What is the advantage of the inverting opamp circuit over non-inverting one?
The inverting configuration is capable of gains less that 1, and can be used as a mixer. Here is a good primer.

http://chrisgammell.com/2008/08/02/how-does-an-op-amp-work-part-1/

I don't know exactly why (anyone feel free to chime in), but the fact that negative feedback is holding the negative input terminal at 0v means that node is a proper place to sum currents, making the mixer circuit viable (although inverting). Op amps are also cheap and come in packages with more than one, so you can usually just invert something again if it's "upside down"