You're not really asking the right questions.
To make the most of the 970 with a 6300 you need to be pushing 4/2/4.3ghz.
That's going to require a $70-80 motherboard & a $25 aftermarket cooler.
The i3 will be fine with a $50 motherboard & the stock cooler.
You have to take that extra expense into account.
I'm not adverse to a 6300 with a 970 as I run one myself but you can't skimp on other components quality wise.