A keyword search for "termination clause" misses a contract that says "the agreement may be ended by either party." Semantic search does not, because it compares meaning rather than matching text strings, which is the difference that makes it useful for anything beyond simple, well-worded lookups.
How it actually works underneath
Text is converted into a vector embedding, a numerical representation of its meaning, and a query is compared against a database of these vectors to find the closest matches, not the closest spelling. The same underlying data often needs to exist in two forms: structured storage for filtering and calculation, and vector storage for this kind of semantic retrieval, and systems that only build one of the two tend to miss cases the other would have caught.
Where it shows up in practice
Semantic search is the retrieval half of what most people call RAG (retrieval-augmented generation), the step that finds the relevant passage before a model is asked to answer a question about it. It is also what makes an internal knowledge base or a document archive actually searchable by someone who does not know the exact terminology the original author used.