A couple of years ago ,when the conventional wisdom was that you buy an i3 instead of a Phenom 955 ,tomshardware did consecutive $500 system builder marathons using those two processors and a Radeon 6870
The phenom nuked the intel pretty much everywhere except games . In gaming it needed to be overclocked to 965 speeds to match the i3 . Current i3's are a little faster , but you can also add a couple of hundred MHz more OC to the Phenom
Mostly though the difference shows in online gaming . The dual core tanks .
Make sure you use an AMD motherboard with a 9XX chipset . probably 970 since you are on a budget . That gives you the option to upgrade to better faster FX processors for at least one more generation .
By contrast the socket 1155 i3 dies in a few months when intel release haswell