The Google Nexus 7 has a higher resolution screen and a faster processor (Snapdragon S4, opposed to the Samsung Note's A7).
However, the Samsung Galaxy Note 3 Neo is expandable with a micro SD card, while the Nexus 7 is not, and the Galaxy Note has a better camera. Additionally, the processor of the Samsung Galaxy Note is only weaker because Samsung prioritized battery life; it's technically a hexa-core, but can run at most 4 cores at a time, and switches between its dual core module and quad core module as battery life and performance demands. This gives it a bit of latency and makes it noticeably slower than the Nexus 7's Snapdragon, but also improves battery life significantly.
I would recommend the Nexus 7 to you, since it holds the advantage in 3 of your 4 points (gaming, reading, watching movies) due to the quicker processor and higher resolution screen. However, if you're going to take tablet gaming seriously, it's recommended to have a Tegra 4 or Tegra K1 processor anyway (Such as the EVGA Tegra Note or the HP Slate), as those typically support more effects. Really, the gaming performance between the Nexus 7 and Galaxy Note are going to be similar in that regard, though the Nexus 7 will fare a bit better. If you were to expand the important points to battery life and max storage, then the Galaxy Note would be more competitive, but based on the points you listed, the Nexus 7 would be the better choice.
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