Now that I have both systems, I feel it is fair to submit a comparison.
My Nexus 8 (SG-8R20) is the stock hub on my Trek L200, with a roller brake mounted on it and shifted by the combined brake/grip shifter unit (SB-8S20).
I've also ridden a Shimano Alfine bike at my LBS shifted by a bar-end shifter/travel agent combo.
My SRAM S7 is the drum brake unit, run with a triple in the front and tensioned by a short cage Suntour Superbe Pro derailleur.
For comparison, I've ridden Sturmey AW hubs for years.
The Sturmey AW is more efficient than all of the above in all gears. The most inefficient gear on the AW (Gear 3) is more efficient than Direct Drive on the Shimano Nexus (Gear 5).
The SRAM S7 is nearly as efficient as the Sturmey AW in gears 3-5, with efficiency in gears 1 and 7 noticeably reduced. With my particular configuration, with the triple in the front, I have a wide band of efficient gears, with emergency bailout gears for climbing and descent gears when necessary.
The main problem with the Nexus 8 is that its least efficient gear is Gear 4 (based on subjective tests, around 80% efficient). The Nexus 8 is a derivative of the Nexus 4, which had 4 gears with 1st gear as direct drive. On top of the relatively efficient Nexus 4, shimano added another epiciclic gear. Thus, gears 1-4 have another 5-7% loss on top of the top gear losses. Gear efficiency, from lowest to highest seems to be: 4, 8, 3, 7, 2, 6, 1, 5. The main advantage to this design is that the primary climing gears are more efficient than those on the Sram S7. The main disadvantage is that some of the most used gears (3,4) are rather inefficient. Most of the time, I even prefer to avoid gear 4 because it is so draggy.
For comparision, some of these losses have been mitigated with the Alfine hub. Needle bearings, shared with Nexus Premium, reduced losses in all gears. The roller clutch in the Alfine further improves efficiency. Nevertheless, efficiency is still lower than Sram S7 in gears 3-4, and slightly lower overall.
2. Cable connection
All 3 speed designs have much simpler cable/shifter systems, so I'll leave them out here.
The Shimano Nexus cable is easier to setup on a new bike. Cut cable to length, install nut, adjust. The installation of the cam nut into the shift mechanism inboard of the chainstay is a bit of a pain without tools.
The Sram S7 Shifter/Clickbox combo comes pre-cabled and set up with an initial length of 1.65m. Once set up, the Clickbox and its quick release is a joy to use compared to the Nexus cable nut. However, the clickbox sits exposed outside the chainstay, making it vulnerable to impact. This is slightly mitigated by the included bumpguard for the clickbox. Cutting the shift cable to length is much more of a pain with the Clickbox than with the Nexus cable.
3. Shift under load
No question here. Nexus is significantly better than the others at shifting under load, with Alfine further improved with its roller clutch. The only thing to watch for is the 4-5 shift, where it shifts 2 sets of epicycles simultaneously (soft pedal).
Sram S7, while more efficient in gear, requires intentional soft pedaling while shifting. Pretty similar to Sturmey AW.
Of course the 3 speeds are lighter.
Between the Sram S7 Drum and the Nexus w/Roller Brake, the S7 is about 400g lighter.
5. Braking performance
The roller brake on the Nexus is stronger than the drum on the S7. However, the roller brake is more affected by weather. During winter rides, the brake mechanism had a tendency to freeze in place.
6. Shift gaps
The S7 has a very consistent shift gap, with smaller shift gaps at 1-2 and 6-7. The Nexus generally has good shift gaps, except the annoying 5-6 shift, with its 28% shift gap.
7. Chaincase compatability
The inboard shifting mechanism on the Nexus lends itself better to chaincase use. From the research I did briefly, none of the Hesling or De Woerd chaincases typical on dutch bike are compatible with the Clickbox on the Sram S7. The Hesling Sabre on my Trek L200 could be improvised to work by removing the right sprocket cover.
The Nexus is dead-quiet, except for some very slight ticking in gear 3. Freewheeling is also incredibly quiet, and completely silent in some gears.
The Sram S7 is even louder than the Sturmey AW. So much amazing mechanical ticking noise, but it may be annoying to some.
The Nexus gripshift with integrated brake lever is an improvement over the regular gripshift. The Sram S7 shifter has a lighter shift, with a more definite click. Matter of preference, but I prefer the S7 shifter.
Overall general recommendation:
Get Nexus 8/Alfine for Hilly terrain, where gears 3-4 will be used less and where its efficient climbing gears (1-2) are very useful.
Get Sram S7 for flatter terrain, where climbs are more rare, where you can take advantage of its more efficient mid-band.