What I really believe what's happening is the following:
PS4 is not really installing the games initially, but rather copying the essential components of the game to launch the game. From that point on, they continue "caching" starting from the most frequently used components to the least. Then when you first load a campaign, it will be relatively slower since chances that not all of the parts were cached, forcing some parts to be read from the disc. The good thing is that this is really a well executed / streamlined process, and further loading will be faster as they'll be entirely cached.
On XB1, however, they are just basically installing the game; stage by stage. This is why it takes this much time. So the stages that are installed are instantly available, and load slightly faster than PS4 as XB1 reads the entire data from the hard drive unlike PS4 which reads parts of data from the Bray disc until the installation process is complete.On XB1, this is an afterthought. MS didn't originally plan such a feature, and they are merely trying to "response" to the streamlined process on PS4. Things may improve later though.
That's exactly what's happening. The goal is to overcome the limitations of ones internet conneciton, blu-ray speeds, and the speeds of the HDD or SSD.
One way to get around this is to install just the parts you need to start the game while the rest is insatlling in the back ground. Sony announced these features a long time ago and are delivering on this promise. Microsoft are playing catchup. Either way, I don't think that this means that the XB1 will beat the PS4's performance even if they add the features later. The reason is AMD Mantel and just overall better compute power in the PS4.