Your electricity bill would change, since browsing the internet does not put your card at full load. Just to give you an idea, my 660Ti rig drew 140ish watts when idling (i.e. browsing internet, very light usage) and up to 320ish if i remember correctly when mining. At least where I live, that's almost 2 cents more per hour
With the 660Ti, I was able to get about 55kh/s, so you could probably pull a little more from the 670. I'd say if you have free energy, you might as well, but otherwise the earnings after energy cost would probably not be worth it.
Do what I did: sell your nvidia and buy a radeon (i went from 660ti (55kh) to 7950 (500kh, could go higher but i'm hitting temperature issues))