Since the R5 reviews have come out, the AT one provides some relevant information since it has a rather good mix of comparing an i5 7600 against an R5 1600X using both the GTX 1060 and RX 480 for both. The game selection is somewhat limited, but it's an okay mix of newer and older.
Civ VI (1060)
R5 1600X: 63.64
i5 7600: 48.21
Civ VI (480)
R5 1600X: 69.14
i5 7600: 50.01
Shadow of Mordor (1060)
R5 1600X: 88.59
i5 7600: 93.32
Shadow of Mordor (480)
R5 1600X: 85.55
i5 7600: 92.15
RotTR (1060)
R5 1600X: 86.48
i5 7500*: 89.76
RotTR (480)
R5 1600X: 80.63
i5 7600*: 81.50
Rocket League (1060)
R5 1600X: 64.15
i5 7600: 105.60
Rocket League (480)
R5 1600X: 182.59
i5 7600: 181.93
GTA V (1060)
R5 1600X: 71.16
i5 7600: 73.20
GTA V(480)
R5 1600X: 57.23
i5 7600: 58.94
To summarize, the 1060 wins 3/5 games and the 480 wins 2/5. The i5 7600 wins 3/5 games and the R5 1600X wins 2/5. There are some weird results, such as RotTR not having the 7600 in the charts for the GTX 1060 (I used the 7500 score instead) and even stranger the i5 7400 with the RX 480 has an average FPS of 91.11 which is technically the best for that game, but seems like such an outlier on the chart that I'm not going to count it since it has to be an aberration or a type when building the chart. Another oddity is the GTX 1060 score with the R5 1600X in Rocket League which is just abysmal, but this could just be a weird driver quirk that can get fixed pretty easily, but even the performance isn't that good on the i5 7600, so I just think NVidia doesn't care much about older games or optimizing drivers for them.
The RX 1600X generally hangs with the i5 7600 typically lagging no more than a few frames behind when it loses. However, in Civ VI it trounces the i5 pretty heavily, which probably shows future performance in games that are thread heavy and use a lot of CPU.
The 1060 and 480 are also similarly close in most cases, usually within a few frames of each other for whichever processor is best. The 1060 gets destroyed in Rocket League, but I think that just comes down to AMD bothering to optimize their drivers for the game and NVidia not caring. It doesn't matter in your case anyhow since both are going to pull acceptable frame rates. The opposite happens in GTA V where the 1060 has a sizable lead over the 480.
I'd be more inclined to go with the R5 1600X (or perhaps the 1600 if it can OC to similar levels) as its going to be more future proof. Also if you have a second monitor and have netflix or something else running in a browser while gaming, the 1600X will handle that more gracefully than the 7600 will. Another thing to consider is that the review lists the 99th percentile FPS as well and even when the 7600 has better overall FPS, the 1600X typically has better results here. I'd give the 1060 an edge on the GPU side, at least for the games tested. If the 580 comes out soon and maintains the same price as the 480, this might change and give the AMD card a small edge on the whole.