It's bizarre that algolia is a thing. Elasticsearch is good and free.
A river was just a mapping from a database to ES. For example, you could search CouchDB with ES with the proper river set up.
ES actually started with Rivers being part of Elastic but since deprecated it. You will still find people talking about "rivers" though as a description for however they are updating the index.
Algolia provides things that elasticsearch do not, such as being hosted, being extremely simple to setup and having a default web interface for anyone to use.
algolia does that out of the box.
I wish Yahoo would have open sourced some of it's internal search tools. :( They weren't always simple to use, but they were stable.