Which build is better for the money?

Solution
It's not hard to make your computer, but if your mother says you're not allowed, you're not allowed.
If you're asking which the better build is, its the second one because of the stronger GPU, but that doesn't mean the rest of the build is trash.

Your CPU is alright and GPU too, if you're planning on playing Minecraft and League of Legends. (my old 5570 could run both)

But beware, your Power supply on both builds are not very reliable, and it could die taking your whole system with it.
To save some money you can cut out the DVD reader, and the CPU fan. (use the stock one and don't OC.)
Also, get rid of that trash Power Supply.
This would save around $58.

Given you want to keep the same budget, (I'm using the $500 one) I would get...

AgentTran

Honorable
Jan 21, 2014
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It's not hard to make your computer, but if your mother says you're not allowed, you're not allowed.
If you're asking which the better build is, its the second one because of the stronger GPU, but that doesn't mean the rest of the build is trash.

Your CPU is alright and GPU too, if you're planning on playing Minecraft and League of Legends. (my old 5570 could run both)

But beware, your Power supply on both builds are not very reliable, and it could die taking your whole system with it.
To save some money you can cut out the DVD reader, and the CPU fan. (use the stock one and don't OC.)
Also, get rid of that trash Power Supply.
This would save around $58.

Given you want to keep the same budget, (I'm using the $500 one) I would get rid of that PSU and replace it with a little more expensive but reliable Corsair CX430, and then with whatever may be left, try to upgrade your graphics card.

PCPartPicker part list / Price breakdown by merchant / Benchmarks

Power Supply: Corsair Builder 430W 80+ Bronze Certified ATX Power Supply ($39.99 @ Newegg)
Total: $39.99
(Prices include shipping, taxes, and discounts when available.)
(Generated by PCPartPicker 2014-03-09 13:45 EDT-0400)

That PSU takes up $40, leaving you with $18 to add to your GPU budget.
Your GPU right now is $70 so we can add that to $18 and remove it.

Your new budget is about $90. Here's the best I could find:
This is a good card and will run most if not all your games you're trying to play at 1080p, but if you're going after BF4 and such, stick to low resolution.

PCPartPicker part list / Price breakdown by merchant / Benchmarks

Video Card: Sapphire Radeon HD 7770 GHz Edition 1GB Video Card ($89.98 @ OutletPC)
Total: $89.98
(Prices include shipping, taxes, and discounts when available.)
(Generated by PCPartPicker 2014-03-09 13:48 EDT-0400)

Hope I helped.
 
Solution
Slightly higher, but you could save money and stick with the video card you had picked out. However, this offers a lot better performance, and has an upgrade path too.

PCPartPicker part list / Price breakdown by merchant / Benchmarks

CPU: Intel Pentium G2120 3.1GHz Dual-Core Processor ($69.98 @ SuperBiiz)
Motherboard: Asus P8H77-M LE Micro ATX LGA1155 Motherboard ($49.99 @ Newegg)
Memory: Crucial Ballistix Sport XT 8GB (2 x 4GB) DDR3-1600 Memory ($69.99 @ Newegg)
Storage: Seagate Barracuda 1TB 3.5" 7200RPM Internal Hard Drive ($54.99 @ NCIX US)
Video Card: XFX Radeon R7 260X 2GB Video Card ($125.98 @ SuperBiiz)
Case: Antec NSK4100 ATX Mid Tower Case ($19.99 @ NCIX US)
Power Supply: Corsair Builder 430W 80+ Bronze Certified ATX Power Supply ($19.99 @ Newegg)
Optical Drive: LG GH24NSB0 DVD/CD Writer ($15.99 @ Microcenter)
Operating System: Microsoft Windows 7 Home Premium SP1 (OEM) (64-bit) ($84.98 @ OutletPC)
Total: $511.88
(Prices include shipping, taxes, and discounts when available.)
(Generated by PCPartPicker 2014-03-09 13:55 EDT-0400)

Budget Cpu, but the motherboard allows you to toss in a 3350p quad core cpu down the road. That would offer excellent gaming performance for the price. You aren't overclocking, don't worry about adding in a cooler.

On the prices for the HDD, don't bother getting anything less than 1Tb. Just not worth it. Added the lowest quality budget Psu I'd go with. Under no circumstances should you ever buy a Diablotek Psu. Win7 is cheapest right now, save the money and get it, although, for 5 dollars more you can get Win8.1. Not a bad idea either, I've been using it for a few months and after some easy tweaks it feels the same to me as Win7 did.

One of the best budget gaming video cards out. 120 dollars is an amazing price for this level of performance. Ram was the cheapest 1.5v 2x4gb set I found. Nothing wrong with the brand(they've been around for a long time) and it's not worth trying to save a couple of dollars getting a 1333mhz set.

I'd work on your mom a bit and show her videos on how easy it is to build a Pc. It is seriously easy as long as you take a little care, and it will save you quite a bit of money. Have her supervise so she can learn as well. Good skill to have so you don't pay a shop or megastore 50 dollars to swap out a HDD.