Well, the Wikipedia article mentioned in the comments is quite theoretic. It starts with terms such as knowledge that need a few explanations.
I take a deep breath and try to explain a few things...
1. What exactly is a Semantic Wiki?
To make things simple, you probably know what is a Wiki. The point is probably about semantics, you have an exchange about this, but I try here to summarize. According to the Wikipedia article on Semantics, this is about relationships between words.
The current Web is largely syntactic: a human (or system) needs to know the syntax of a language (including grammar) of a page in order to capture the meaning of the words. This is why I can't personally understand a page in Chinese or Arabian but will guess things in Dutch or Danish: these languages are European, will have a grammar similar and an etymology close to the languages I know.
In 2001, Tim Berners-Lee (father of the Web) et al. introduced in a seminal article the idea that the Web would become very interesting if page terms where exposed on the semantic side as a complement to syntax informations. This is the concept of the semantic Web. Such an organization of data would have several advantages including predicate calculus (also called first-order-logic -> that article is succinct) and internationalization. An important element in predicate calculus is about data quality.
The semantic topic is really huge: if you are really interested then you should read a book about it.
A semantic Wiki is simply a Wiki that takes advantage of semantics rather than syntax. It has the advantage of a Wiki: the collaborative aspect can be precious for data quality. Note: you have several standards for semantic informations (RDF/OWL, Microdata/RDFA/JSON-LD).
2. What is the purpose of it?
At first glance, the main interest of the semantic Web is for search engines: Web pages would be easier to crawl with semantic informations. Web crawler are in fact the main applications processing Web pages.
With some tooling (e.g. an RDF parser), a semantically-structured Wiki could also be easier to process by third-party software than a purely syntactic Wiki. The client could easily build its own knowledge base.
If you want to understand the deep interest of such a system, think about IBM Watson. But for such a result, a huge infrastructure is needed.
Using a semantic Wiki has the advantages of a Wiki. Think about data quality: for Watson, IBM analyzed the content of Wikipedia - they talk about the UIMA architecture for that analyze. In the fourth part are a few words about DBPedia which is a project analyzing the semantics of Wikipedia articles.
But few non-semantic wikis have the quality of Wikipedia.
3. Why and when we use it?
Why ? You have several reasons for using a semantic system. The most obvious use of a semantic system is internationalization. Take a look at the Unesco site in English, in Chinese, in Arabian: three different sites with a pure syntactic solution. There are also many other advantages related to deduction (the predicate calculus topic). An example of a simple deduction application for search engines is Google Knowledge Graph (simple is here a blur point about structured data impacting the functionalities, not the project itself: in a few years it could become something really impressive). Most works on the topic are academic. There is a need for a huge infrastructure to have an effective system. And first of all, you need to understand very well the question of the semantic Web.
For the when part, I would say when your project is backed by a large organization and when you are really familiar with the topic. Another case for using semantic informations would be to help Web crawling. But as far as I know, if Google loads currently Microdata/RDFA/JSON-LD, this has a very light impact on Web ranking.
4. Is it a PHP Framework or something like that?
It would be difficult for me to advise you about a PHP solution.
The Ace Wiki is a good example. The linked page gives access to a geography Wiki which is a sample and the GitHub account of the project. Unfortunately for you, it is in Java.
Semantic MediaWiki is an extension of MediaWiki to enrich pages with semantic informations. It provides a SPARQL interface (the SQL for knowledge bases) and an RDF export. These two functions suggest that the primary data storage is graph-based (I have no experience of this).
Don't be mistaken: you will cross DBPedia but this is NOT a semantic wiki. This is a semantic form of Wikipedia. However, this is quite an important knowledge base.
