Overall, for relatively simple searches, PostgreSQL is good enough, which should probably be your case.
Elasticsearch becomes really relevant when you need more complicated search features, such as complex marriage with the searched terms or using different parsers. If you have a very large volume of searches, elasticsearch also behaves better than PostgreSQL (it uses less hardware for the same result).
As an example, a feature I use on the job and I believe would be difficult to implement with PostgreSQL is the ability to index synonyms for words . Another is the quick possibility to find similar documents or suggest searches for your user.
These features greatly enrich the search for an application and are simple to implement using elasticsearch.