It's not much about benefits that the Xbox One has over the 360. It's more about how the "newer" hardware that developers can take advantage of to make games more complex. If you ask me about benefits between the Xbox One and 360, I would say there is none as they are both game consoles (you can play games on either one, and they both have Kinect). Watching TV on the Xbox One is not much different from streaming on the Xbox 360.
Both are very similar, however, it is the new hardware in the Xbox One that provides it the "new" edge when compared to the 360.
Similar to a low-medium end gaming PC, the new Xbox One is running an 8-core AMD CPU with 8GB of RAM, an AMD Graphics GPU, and comes with Kinect 2.0. Compared to the Xbox 360, this is one hell of an improvement, however, you can see that I put the word 'new' in quotations to describe the Xbox One simply because the hardware used in the Xbox One is still not comparable (and never will be comparable) to PC hardware.
Now, in regards to this new hardware, we can see a substantial increase in graphical fidelity as well as DirectX 11.2 support that developers can take advantage of (although I doubt developers will use this, simply because it is only available on the Xbox One and Windows 8.1 and up). The main improvement will be in the game graphics and not so much of anything else; the next generation of games is more focused on next-gen graphics than anything. Surely, with the increase in graphical settings, the size of the game will also be much larger than Xbox 360 games.
Well, I am not sure if this answers your question about Xbox One vs 360 benefits or not, but hopefully it will shed some light about "how much better it really is compared to the 360." As with all forms of technology, think of the Xbox One as an innovation based upon the 360 with hardware improvements.