With Vale Server, you can use Vale CLI’s style-checking capabilities in editors such as Visual Studio Code, Atom*, Sublime Text, Google Docs, and Chrome.
On your system, you run Vale CLI and Vale Server. In your editor, you run a Vale client that connects to Vale Server and displays style feedback while you’re editing the content.

For example, without using Vale Server, if you run Vale CLI and use the Google style to check some newsletter content in a markdown file, the output looks like this:

With Vale Server, checking the same content in Microsoft Visual Studio Code looks like this:

Hovering my mouse pointer over line 8 displays the same warning message about sentence-style capitalization. Unlike the CLI, this message in the editor gives me a link to docs for the Google.Headings style rule. And if the issue has an automatic “quick fix”, I can click a button to make that change.
In terms of writing and editing, that’s how Vale CLI and Vale Server are different! The other differences have to do with how you configure and administer each one.
Footnote: I haven’t been able to get Atom to work with Vale Server.