Ok, What I remember reading about at around the announcement time - and I do think the logic makes sense here - was that they had decided that the cost was too prohibitive to go with 8gb of GDDR5 because of the memory target they had. So they decided to build in some eSRAM on the die for three reasons.
1) The speed and location of the memory would makeup for a lot of bandwidth loss when trading data between the CPU and GPU cores. The PS4 CPU and GPU have to access their data over a bus capable of supporting the GDDR5 (more chips = more money), while the eSRAM on die doesn't need separate control hardware, because direct access.
2) Because they were building it on die, the cost of the memory would go down over time due to advances in production (Think going from 45nm to 22nm, less heat and less power, more chips per die = lower cost per chip).
3) It was more or less a proven technology at the time. Microsoft, not wanting to repeat the mistakes of the last generation, were more careful in their hardware design leading to longer hardware change iterations. Meaning once they had selected an architecture, they would be unlikely to change it.
In general, integrated components - while initially more expensive to produce - get cheaper over time. The "gamble" they made was that GDDR5 was not going to come down in price in time for them to be able to make their launch window. Sony was "lucky" in that even though they had designed the system to use (for sake of argument) 4x 1GB GDDR5 chips, these 2GB chips came out that were at an acceptable price point and were a pin-for-pin replacement. (Take the old out, pop the new ones in, BAM, double the memory).
EDIT:
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By putting the ram on the same chip as everything else they make the design much more flexible. Not sure if it beats having a large pool of very fast ram, but I wouldn't say it was a bad design either...
Yep, and most of the reports at the time were that even though the XONE's lower memory bandwith, the display targets that both consoles are trying to meet could be satisfied with even lower memory bandwith
EDIT: Woops!
Found that Eurogamer article that points out the bandwith: eSRAM 206GB/s, GDDR5 176GB/s, DDR3 68GB/s
Think of the XB1 as a suped-up Box truck working with a train vs a fleet of 18 wheelers. Both will get your data there, PS4 is "faster" when you consider the average over time of data transfer, but the XB1 is faster but only with a few MBs of data, otherwise it relies on its slower DDR3