Layman's terms, obviously taking liberties with the technical terms.
Legacy memory set-up in consoles:
System RAM + Video RAM.
System RAM is accessed by the CPU, VRAM accessed by the GPU.
Current gen memory set-up:
Xbox 360 came along and had unified memory + a tiny amount of embedded memory (eDRAM). One single pool of fast RAM, accessed by both CPU and GPU. The eDRAM very expensive, very fast and is used for the frame buffer (the final image to be displayed on screen). Allowed to 'free' AA.
The PS3 has the old style split RAM. 256MB for CPU + 256MB for GPU. The lack of embedded RAM meant that many PS3 games didn't have AA. Weakness of the split memory pool is if you games was graphically intensive, you were limited to the VRAM and couldn't use the system RAM. This was the main bottleneck in open world games, compared to the 360 versions.
Next gen memory:
PS4 has a unified pool of ultra fast GRRD5 RAM. This is used for everything, and access by both the CPU and GPU, for system, graphics and frame buffer. The speed of the RAM negates the need for embedded RAM. Devs choose how much memory to assign to graphics, AI, physics etc.
Xbox One as unified memory + embedded memory. This differs from the PS4 in that the main 8GB of memory is slow, cheap DDR3, which isn't suited for graphical tasks. So the Xbox needs a small chunk of fast embedded RAM (ESRAM).
Both the PS4 and Xbox One are APUs. Their CPU and GPU are on the same chip. In the case of the Xbox, also on the chip is the embedded memory (ESRAM). The ESRAM is huge, so the maximum amount they could responsible fit on the APU was 32MB, which isn't enough for 1080p + MSAA, or 1080p and deferred rendering (e.g. a modern games engine).
Another issue is based on dev comments, devs manually have to fill and flush to/from the ESRAM, rather than it being handled by the APIs, which is a huge ball ache.
The tools will improve over time, which will make optimising use of the ESRAM better. Devs can get around the 32MB size being unsuited for 1080p + effects by tiling and jumping through a bunch of other hoops, which aren't ideal.
The PS4 design is better all around. Gives better results, is more flexible, and is far easier to developer for.