Most likely, your desktop would have to print gold to make up for the difference in your power bill compared to a ChromeBox too. My desktop needs a couple hundred watts; my Google TV box needs closer to 10. Running a full-size desktop where you don't need the power is equivalent to a voluntary subscription fee versus a one-time payment.
I'd be curious to compare your time from power off, to viewing a web page.
They're very different machines, with very different goals. If you compare the components, then yes, it probably doesn't make any sense for you to get a Chromebox. If you look at the features they both have, then it might make more sense.
If I were buying a machine for my grandmother, I'd buy the Chromebox over your $377 desktop, in a heartbeat.
That being said, it's a 1.9ghz Celeron CPU. That can give negative connotations since Celeron is Intel's cheap line of processors.