Which Processor is better

Hello... The value is with the Intel MB... You will be able to put a i7 into that very same MB in the future.
http://www.cpubenchmark.net/cpu.php?cpu=Intel+Core+i3-4150+%40+3.50GHz
http://www.cpubenchmark.net/cpu_lookup.php?cpu=AMD+FX-6300+Six-Core
The FX is a physical 3 core CPU... the I3 is a physical 2 core CPU.

The i3 is a better CPU, the 6300 single core performance is comparable with the 5 year old i7. Even with 6 cores it is not as good with gaming as the 2 core i3 w/HT (4 cores).

http://cpu.userbenchmark.com/Compare/Intel-Core-i3-4150-vs-AMD-FX-6300/2309vs1555

Most things (applications/games) still use only about 2 cores (plus HT on Intel side), so the i3 generally will do quite well. Now if you had specific uses that needed more cores, than the FX would probably make more sense. I believe Far Cry 4 is requiring 4 cores (probably 2 Intel cores w/Hyperthreading) to run the game, but I know there are workarounds for this. It really depends on what you want to do with the CPU. For gaming the i3 is probably going to be the better option in the short term. Later down the road who know if the 6 core FX is going to be the better option or not. Intel's higher clocked and generally more efficient architecture will usually reap better and faster rewards.
The other thing that you need to keep in mind is the cheaper Mobo's that you can put the i3 in are going to keep your overall cost down over the FX build. Yes you can build on a cheaper FX Mobo (thinking 970 chipset), but sometimes it isn't worth it.

Here is how I see most budget builds go for cost on the CPU and Mobo:

Intel build:
* CPU i3 ~$120ish
* Mobo for non-OC'ing models ~$50-75

Total cost of CPU/Mobo = ~$170-200ish

AMD FX 6300 build:
* CPU ~$100
* Mobo ~$70-100 depending on sale, but usually 970 chipset to keep costs down (990 FX chipset preferred, but usually is >$100 for the Mobo).

Total FX 6300 build = ~$170-200 with 970 chipset

As you can see the costs are going to be similar, but if you want to OC on the FX 6300 (which should be considered) you would probably need at least a ~$30 CPU HSF (CM Hyper 212 EVO I'm looking at you). This would adjust the cost up, which then you would be close to the Intel i5 CPU prices (~$170-190ish) range.

So it will all depend on what you want to do. AMD has decent CPU prices and performance, but Intel has the better single core performance around. So Intel has the upper hand generally and usually is recommended over AMD for most mid-upper CPU options. If you budget is even tighter, the FM2+ based AMD CPU's are another option to consider. They are usually around $60-80 and you can get an FM2+ based Mobo for around $70-90 (88X chipset).

*Edit
Forgot the CPU benchmarks:
FX 6300 vs. i3 3220 - Product Benchmarks
Bioshock Infinite Benchmarks w/single GPU
You can use Anand's bechmarks charts for just about anything that they have tested, so feel free in selecting the options that you are interested in.

Hello... You asked for some Data to be included, and thus is the PASSMARK charts. And that data is uploaded from real people machines. If they are Overclocking them, I'm not sure if the Data can be tillted from stock settings... And Could very well be.
But as a Builder, The next best CPU upgrade options comes with the Intel MB... and an I5 or i7 in your future would be "awesomeness" Computing Power. B )
I build systems with the i3... and I am very impressed with Over all performance with that CPU and HT, and have happy customers... I have no experience with the FX 6300.

Yes generally the CPU is less of a factor on a gaming build when using a dedicated GPU. The FX will OC, assuming you have sufficient cooling, decent PSU, & a Mobo that can handle most OC's. The i3 will not generally have much if any OC'ing done on it, so keep that in mind. The Intel option will allow you to have a much bigger upgrade path, IMHO, than with the AMD system, but going from a 6 core to 8 core on the AMD side won't change things too much as far as gaming is concerned. I would just make sure if you go with the AMD option and plan on doing a CPU upgrade down the line, to make sure you have at least an 990FX chipset to support the extra power required by the FX 83xx CPU.

Personally I would rather use the $ saved on the CPU/Mobo combo (if you get a decent setup of those 2) and spend it more towards a better GPU. This will allow you to play on higher settings for longer or at least allow you to stretch your build out longer, before needing an upgrade.

It is all up to you on what you would like to do. Either option will be a good one, so pick which one you prefer.

Hello... Yes for Graphics/Gaming programs, The GPU $$$ invested will make a Larger difference than between the i3 and fx-6300 choice here. http://www.videocardbenchmark.net/high_end_gpus.html

Hello... Here is a Passmark single thread chart. http://www.cpubenchmark.net/singleThread.html
i3 4130 1980 fx-6300 1413 25% difference.

Yes you are correct.