There's no overlap between the two. Both Vale and Vale Server use prose to do their internal NLP-related work, while LanguageTool does its own.

The Vale Server + LanguageTool integration is done simply through input/output processing: Vale Server passes text (from one of its clients) to a local instance of LanguageTool standalone and then processes the output provided by LanguageTool. In other words, Vale Server doesn't know about the internal logic of LanguageTool and LanguageTool doesn't know about Vale Server's.

It's also important to note that, as described in the documentation, not all of LanguageTool's rules are currently supported. 

Vale Server offers a free trial to allow you to experiment with its various features and integrations prior to purchasing a license. 

Thank you for the clarification. Two follow-up questions

1) The documentation says what rules Vale *can* support from LanguageTool. I know this is redundant, but what are the rules which Vale can't support?

2) Do Vale and Vale Server use the same dictionary of tagged words? Is it this? Is there any documentation on how correct are the tags, or what the tags mean?

1) You can browse the selection of rules here: https://community.languagetool.org/rule/list?lang=en. The ones listed under the categories mentioned in the docs are all that's enabled for now.

2) That's the default dictionary used for spell checking in Vale and Vale Server (it's not related to part-of-speech tagging). You can read about the tagging process here: https://github.com/jdkato/prose#tagging.

Thanks for all the information, regarding this point:

This makes sense to me. However, the website also says:

So then, if a DITA file has a sentence with inline code like `codeph`, `uicontrol`, or `keyref`, how does Vale Server tell LT not to apply any rules on those tagged elements?