At $400 (before tax) you have 2 choices. Both are from HP and have 17.3" screens. They are actually the same laptop but one has a quad core AMD A8-5550m 2.1GHz (3.1GHz Turbo Core) APU with the integrated Radeon HD 8550g graphics core. The other has a dual core Intel Core i3-4000m 2.4GHz CPU with the integrated Intel HD 4600 graphics core.
AMD:
http://www.bestbuy.com/site/pavilion-17-3-laptop-4gb-memory-750gb-hard-drive/2878437.p?id=1219080293705&skuId=2878437&st=categoryid$abcat0502000&cp=1&lp=1
Intel:
http://www.bestbuy.com/site/pavilion-17-3-laptop-4gb-memory-750gb-hard-drive/2854029.p?id=1219079079760&skuId=2854029&st=categoryid$abcat0502000&cp=1&lp=2
The Intel HD 4600 graphics core is more powerful than the Radeon HD 8550g graphics core. There are only two AMD integrated graphics core that are more powerful than Intel HD 4600, but they are found in more expensive AMD laptops; Radeon HD 8610g and the Radoen HD 8650g. That means you will get better game performance with the Intel version of the HP laptop.
On the processor end the AMD A8-5550m is a quad core APU with a base clockspeed of 2.1GHz. Turbo Core is an automatic overclock; while it can reach 3.1GHz that can only be done if only one core is used and if the APU is not running very hot. If 2 or more cores are used then the Turbo Core speed would be at best 2.9GHz. Maybe a little lower. Having 4 core means that 2 cores can be devoted to playing games and 2 cores can be used to record the video.
The Intel core i3-4000m is a dual core CPU running at 2.4GHz. Intel CPUs are very powerful compared to AMD APUs. In fact, despite the fact that it is only a 2.4GHz CPU it would be comparable to an AMD APU running at around 3.0GHz to 3.2GHz. While it only has 2 actual cores, it has something called Hyper Threading (HT). They act like virtual cores / "not real" cores that can help when recoding videos. Games do not make use of HT though so from that point of view it will not help, but Windows background processes does use it and it can assist with video recording.
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So which one to choose....
- The Intel version will give you better game performance because the graphics core is stronger.
- 4 actual cores is better than 2 actual core + 2 "fake core". This generally means that when you are recording video while playing games, the AMD version may give you better performance. With the weaker graphics core since you can devote two core to the game and two cores to recording.
- While Hyper Threading does not make a dual core CPU into a quad core CPU, it can help with handling the background processes and recording video.
If you are strictly playing games, then the Intel 17.3" HP laptop will give you better performance. When playing games and recording, then it becomes debatable. Either one could be better than the other or they may end up being somewhat equal to each other.
Regarding 60 FPS in Minecraft, the answer would be probably no. Especially when recording. These laptops have 1600x900 resolution screens which means lower performance compared to 1366x768 resolution. You can always drop the resolution down to 1366x768 when playing games. Lowering graphics settings can also get you closer to 60 FPS.
I do not play minecraft, but I think in an open area with not many object, you can get 60FPS, but from what I've read jungle areas are especially demanding so if you can get 60FPS in open areas, you may only get 30FPS - 40FPS in the jungle. This is just an educated guess. However, with a limited budget of $400 these two laptops are the best you can hope for.