Connor is right that a more powerful graphics card is better for minecraft (main benefit I know of - allowing vision distance to be larger). But he's wrong about a Bulldozer or quad-core Phenom outperforming an i3-2100. The i3 is a fantastic gaming CPU, and only in a few games does any of the AMD chips equal its performance - and that's assuming you overclock them heavily.
But most games don't care what CPU you have, and minecraft is one of t hem. So really, you should worry less about IPC, cores, and gigahertz, and get whatever is decent for cheap so you can have a better video card. That, by the way, is really the only reason AMD is 'bad' for gaming - Intel CPUs are less expensive when you take into account motherboard/psu/cooling costs. Below is what I'd go for, assuming what you want the computer to do is minecraft + sometimes some other games:i3-2120 - ($110)
Pentium g630 - ($68)
Biostar B75 Mobo - is cheap, works, and even has USB 3.0. Luxury! ($60)
MSI 7870 ($280-$20 rebate)
2x4gb 1333mhz RAM ($35 after promo code)
Antec One - good budget case, front panel usb 3.0 ($50)
Antec Neo Eco 450c - Quality power for cheap. nice. ($35)
Seagate Barracuda 750gb ($60)
http://us.ncix.com/products/?sku=49597&vpn=DRW%2D24B1ST%20Bulk&manufacture=ASUS&promoid=1145Total: $605 w/g630, $585 AMIR
There. The processor is only a dual-core, but it's reasonably fast. It's plenty of power for minecraft and the almost all games, and thus allows you to get a 7870 - a video card with some real power. The 7870 is significantly more powerful than even a 6970 (assuming you managed to unlock the 6950). Plus the overall build is $100 cheaper. Now THAT's better for the price.
If the CPU makes you nervous (it shouldn't), you can still use the extra money left over to get the i3 crossed out above or even an i5-3450 if you don't mind stretching your $700 budget by $20 or so - and the i5-3450 is pretty much incontestably better than any AMD processor for gaming (though not necessarily for other purposes).
Alternatively, if you wanted to spend the whole $700, you could even get a 7950. If it were me, I'd probably get the above but with the i3 ($647). For minecraft/other casual gaming, though, there's really not any reason to get more than the g630.Edited by MisterFred - 8/3/12 at 5:56am