For gaming, the R9 380 should in most titles be the faster card, even for a model with the same amount of VRAM. If the one you are looking at has 4GB of VRAM, while the GTX 960 only has 2GB, then the 380 would likely fare a lot better in modern AAA games, as 2GB generally isn't enough to run many recent titles smoothly. The 960 should draw less power under load though, and in turn put out less heat, so if you have a low-capacity power supply it might be worth considering.
As for getting down to comparisons of one specific brand against another, the Strix line from Asus is generally considered more of a premium brand, while Zotac is generally more of a lower-end brand. Anecdotal comparisons from someone having problems with their card from a particular brand probably don't mean much, as it's possible to find examples of people having problems with any given model of graphics card on the market.