This is my view: Cisco has been strategically acquiring companies to strengthen their portfolio in security in general. One of the largest acquisition is Sourcefire, and mostly their AMP technology. What Cisco is trying to achieve is embedding AMP into its core set of devices/services. Customers are not very keep on seeing more and more boxes and interfaces so it is critical to keep everything nicely and tightly integrated - and this is not something that is done easily or in a week.
AMP Everywhere implies the fact that Cisco is working on embedding AMP technology (and it's detection and retrospection capabilities) into ASA, CWS and others. As such it is also term used for marketing purposes.
The power of AMP Everywhere within an organization's security infrastructure is really the power of having more eyes in more places. Malware is dynamic, moves quickly, works together with other malware in the environment, and attacks via multiple attack vectors: email, web, on the endpoint, mobile devices, software vulnerabilities, the list goes on. To truly protect your organization from threats, you need to get maximum visibility and control into as many attack vectors as possible.
Single AMP deployments are powerful on their own. Many organizations are fine with, for instance, deploying AMP at the endpoint or deploying AMP at the network to get the amount of visibility and control they need. However, many choose to deploy AMP simultaneously at multiple security control points to increase security effectiveness for the organization. With AMP in more places, you get more eyes watching more attack vectors, continuously monitoring for malicious behavior across the extended network. AMP also shares intelligence across deployments, meaning if one deployment sees something, all deployments will know about it. This increased visibility and control allows organizations to respond to, detect, and remediate threats faster.
Here are all of the AMP deployment options on one slide: