8350rocks :
The FX8350 is as good as the i5-3570k at anything and better at many things. Yes, you would see something on the order of 20% improvement in nearly everything you do. Yes, it would be worth it. Update your BIOS to newest version and rock the 8350!
First of all the FX8350 is NOT superior to the i5-3570K for gaming. That's been proved many, many times with the exception of TekSyndicate who have a completely skewed analysis and claim they are the "only" ones who know what they're doing (for example, they said the 8350 + GTX670 got 55FPS in Far Cry 3 but only 25FPS with the i5-3570K.
*Having said that, if you buy the i5-3570K you need a new motherboard. Unless you have Windows 8, you'll have to buy a new copy of Windows because pre-Windows 8 OEM copies (you likely don't have FULL) are not transferrable so that's another $100.
Also, your graphics card is okay, but not the best so you will be GPU-bottlenecked anyway in some games.
So I do recommend the FX-8350 if you think it's worth it. That's about $200 assuming it's compatible with your motherboard. If you bought the i5-3570K + Motherboard (then Windows) you're talking over $400 (probably $500 with tax/shipping).
You can also Google benchmarks to see how much of a difference upgrading from the X4 965 actually makes (this will vary a lot). Games which are more CPU-dependant will obviously see the biggest gains.