The 2013 M5 you cite has the BMS tune, downpipes and aftermarket exhaust. Quarter mile at 11.073 seconds at 128.680 mph.
The fastest 2014 E 63 time with tune, downpipes and aftermarket exhaust (AMS mods) in the quarter mile is 10.850 seconds at 129.310 mph: 2014 Mercedes-Benz E63 AMG S AWD Alpha 1/4 mile Drag Racing timeslip specs 0-60 - DragTimes.com
To me that is not that close.
In stock form, in the comparison I cited, Motor Trend clocked the M5 Competition Package from 0-60 at 3.7 seconds and E63 S at 3.4 seconds. From 0-100, the M5 was at 8.1 seconds while the E63 S was at 8.0 seconds. In the quarter mile, the M5 was at 11.9 sec @ 122.2 mph while the E63 S was at 11.6 sec @ 121.8 mph. To me, these results all show that the E63S has better acceleration at least until 100 mph. Clearly, above 100 the M5 will start reeling the E63 S in and passing it due to the greater high speed efficiency of rear wheel drive. But, to me, that is not important because I don't drive at speeds above 100 mph in real world driving. I usually don't exceed 85 or 90 and my average highway cruise is about 70-75.
Moreover, the M5 is very difficult to launch correctly due to so much power just driving the rear wheels. With the E63 S and its all wheel drive, it is point and shoot as it just puts the power down to the ground. Stated differently, the average driver will be much faster accelerating in the E63 S than in the M5.
In terms of braking, Motor Trend found that both cars stopped from 60 to 0 in 106 feet.
In terms of handling, both cars pulled .95 g on the skidpad and had the same time through the figure 8: 24.3 seconds.
Of course, in corners, the M5 rolls less than the E63 S which will roll more. I do love me some flat cornering.
Here is the link to the MT stats:
2014 Mercedes-Benz E63 AMG S vs. BMW M5 Competition Pack - Motor Trend Page 2
Again, to me they are both great cars. I would be happy with either one. I chose the E63 S because of the greater power, tuning potential, acceleration and winter traction with its AWD and because of its amazing engine and exhaust noise which, to me, the M5 does not compete with.
I still think the M5 is the better looking car both inside and outside as well as having better electronics, stereo and transmission.