First, for your research you will likely find this post about the theory of moderation on StackExchange very useful as it describes the concept of how moderation is intended to work.
As for the specific question of community moderator powers vs 4k/20k rep users (beta sites only require 4k for the 20k privs and 2k for the 10k privs), it is fairly similar, but a few key differences. The biggest one is that while high reputation users can vote to close, delete, or even migrate posts in certain cases, some criteria must be met and the action must be confirmed by multiple users (generally 5, sometimes 3). You can see more details about the specifics of high rep user privileges by browsing the privileges page and clicking through the different levels of permissions.
On the other hand, diamond moderators have similar permissions, but without the restrictions or need for validation of the decision. Close, Delete and Migrate actions all take effect immediately. Close votes can be overridden by the community (via a reopen vote), however delete votes cannot. Diamond mods can also vote to close or reopen a post more than once (normally once you vote to close a post once, you can't vote to close it again, even if it is re-opened. Similarly, once you vote to re-open a post you can't vote to re-open again, even if it is closed. This is to limit the length of open/close wars.) Additionally, diamond mods are able to migrate to any site on the network at will, not only a selection of up to 5 pre-approved migration sites.
Diamond mods also have some extra tools designed to help them with their special duties such as verifying spam posts and other violations of terms of service that aren't needed by normal users. For example, diamond mods are able to destroy accounts used by spammers and add their connection information to lists of spammers to make it harder for them to keep spamming. They are also able to issue suspensions to users who are violating site policies either severely or regularly. They are also the only ones to see the "Flag for Moderator Attention" flag queue, which can be used for more sensitive issues that someone doesn't want to have the community handle.
Even for diamond mods, some actions require working together behind the scenes.
The general idea is that diamond mods represent the will of the community and are users that have a solid understanding of what the view of the community is. In situations where the view of the community isn't clear, then moderators should do their best to let the community decide, however, in cases where something is clearly against community norms or policies, they are able to step in and resolve the issue quickly to avoid problems and reduce noise.