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Is an Intel i5 better than an AMD FX-6300?

Lower power required, more efficient instructions per clock. Although the 6300 has six logical cores it doesn't have six full sets of independent hardware. It has 3 sets of hardware and then they split into to two cores each. Since the two cores share resources you don''t get a 2X improvement. So you have a full quad core Intel vs a 3.5 core AMD. Depending on the i5 part the Intel CPU has more on chip cache.
 
All the answers above are good and relevant.

Define "better"

If your usage is multithreaded apps, then the 6 threads of a FX-6300 is a very decent budget value.

If your usage is for gaming, then that is a different story.

Most games depend on one or two fast cores. Yes, there are exceptions, FSX for example. The extra 4 threads of the FX-6300 go largely unused.

The intel haswell cores are perhaps 30% more efficient per clock than the FX cores.

 
A FX-6300 is the only amd cpu that makes tom's best gaming cpu's for the money list:
http://www.tomshardware.com/reviews/gaming-cpu-review-overclock,3106-2.html

If you ever plan on a cpu upgrade, all the other higher and lower priced options will use a lga1150 motherboard, allowing all sorts of cpu upgrades.
Here is an older comparison which includes a FX-6300. Note that the intel i3 chips are older tech and still do better.
http://www.tomshardware.com/reviews/gaming-processor-frame-rate-performance,3427-9.html