There is no proof yet that X1 is on a completely different Feature Level then PS4 therefore what Shifty writes is correct, if there no difference hardware wise then any difference can be made up by Sony just by developing them, should they choose to. However, this is the part where my agreement begins to diverge; the notion that the API changes will benefit both systems symmetrically.
With the introduction of DX12, it will undoubtedly change the method in which games are programmed; as such bottlenecks on the system are expected to move. Past performance in games will not be good indicators for the future, and past bottlenecks may not be future bottlenecks. DX12 shows a strong movement towards reducing CPU overhead whilst simultaneously increasing GPU efficiency/CU saturation by reducing the amount of time the CUs will stay idle waiting for work to arrive. However GPU efficiency comes at a cost, it is not free, improving GPU efficiency has three additional requirements, the first is increased sustained bandwidth to feed the CUs to do more work, the second is the heat/longevity of the silicon, and lastly the increased power requirement. How PS4 and XO approach these limitations is actually different, as such since their approaches are different they cannot equally gain the same amount. My position is that Xbox One is better designed in these aspects for DX12 than PS4 is, thus has more to gain moving to DX12 than PS4 would. [Please note this does not mean I expect X1 to out perform PS4, but I do believe the gap will shrink]
When looking at available bandwidth, this AMD paper [
http://research.cs.wisc.edu/multifacet/papers/micro13_hsc.pdf] specifically notes
Page 4: Specifically, for a GPU composed of 32 CUs, we found that 700 GB/s eliminated the memory bandwidth bottleneck for all our workloads.
Doing some math will indicate that @ 853 Mhz at 12 CUs, the required bandwidth to remove the bandwidth bottleneck for CUs will be approximately ~240 GB/s. This happens to be approximately the total complete bandwidth of Xbox One (192 GB/s + 67GB/s). When looking at PS4 we have a max theoretical of 176GB/s, and as such, bandwidth will become a bottleneck for all 18CUs. Math will show that there is only enough bandwidth to fully saturate 9CUs. From a design perspective Xbox One is better built for sustained high bandwidth work, the less idle time the more performance you can obtain from the Xbox. So much so that in it's perfect impossible form it can fully saturate 12CUs vs PS4s 9 CUs. This is not a strong argument to compare performance between the two systems, but this is a strong argument on how their systems are designed for the new API.
The second factor is thermal. The GPU efficiency is bound to increase as idle time for the CUs drop, this also has a side effect of increased thermals. Xbox One reportedly had very loud fans for developer units because they did not have sensors ready so they ran the fans at 100%. To date no game has ever caused the Xbox to exhibit more noise such that it becomes more audible. A simple look at the hardware for Xbox One and we see an ethusiast level cooler for a SoC that requires significantly less power than its PC counterparts.
The Xbox One does use less energy than the PS4 when it's playing games (112 watts versus 137 watts) or streaming videos (74 watts versus 89 watts). Polygon,
http://www.nrdc.org/media/2014/140516.asp
The PS4 does not have such a cooler, today is exhibits increased fan noise when playing specific games. The question becomes, how well tested is the hardware for this level of GPU efficiency?
Can it take the heat? Excessive heat and voltage will cause the lifespan of a chip to die, so are the chips ready for 7 years of torture? I'm confident that XBox One can its heatsink cooler cools only the SOC and a majoirty of its bandwidth comes from esram which is centralized on the SOC, I'm not sure if the PS4 can, its bandwidth is off SOC, thus increased pressure will cause RAM to heat up as well (there are no heat sinks on GDDR5 like on PC parts). If you like to overclock you know Prime95, and when running Prime95 if stability is an issue you need to both underclock, increase voltage, or decrease speeds on the bus, on RAM, essentially where ever the weakness exists. In this scenario, I believe once again Xbox One is better designed for the high saturation games we should see coming with DX12.
Lastly the final factor is power. Not only does Xbox One use less power than PS4, but it also has a large dedicated external power brick vs the internal power supply of the PS4. Overtaxed power supplies are a less common cause to crashing in Prime95 but it still can be a factor. There are just very few programs that ever push your chips to 100% but power supplies that are pushed to its limits over sustained periods of time the increase heat in the power supply can cause voltage drops resulting in crashes and general instability. The Xbox One has an external brick with it's own dedicated fan vs PS4 internal with its shared SoC fan. So once again from a power perspective we see that Xbox One is ready for DX12, PS4 is not as prepared.
When we consider these points together, it is clear to me at least that MS has designed the console to be ready for DX12. They knew it was coming, they knew what it would it do to the silicon. All areas of the console have been beefed up to prepare for it and these design choices reflect that DX12 was meant to be the endstate for Xbox One; ultimately we should see performance of the console improve greatly as engines move towards DX12 based design. For PS4, I don't see a console that is ready for the additional burden. Sony could choose to implement all the features found in DX12 but they must determine whether or not the hardware is suited for it, for the remaining lifespan of the console. Will their units begin to fail because the way newer games are designed. If so, then the way to control it is to not allow full access to these features, optimize the API only as much as their console can take. The newer games will still drive PS4 to its limits, but the limits of what Sony wants (whatever that may be).
edit: tl;dr; You can't put something into the SDK and then later take it out at another date. If they had never designed the console with this particular level of stress in mind, then they need to be very thorough on testing on what to enable. Sony doesn't want to be in a situation where all the Gen 1 consoles start dying by year 4-5. This isn't applicable to Xbox One however. They are fully aligned for this, and have stress tested accordingly.