Disclaimer: I'm a developer at Algolia.

IMHO the two main advantages in favor of Algolia, are the sane defaults for relevancy and speed and the fact that the service is hosted and can grow with your business without having dedicated engineers to manage both the configuration and the infrastructure.

Also, on top of the Algolia services per se (search, analytics, recommendation, etc.), we're providing a lot of backend and frontend libraries which one would otherwise need to reimplement when using an elastic- or Solr-based implementations.

Search is hard to get right and the cost of Algolia is negligible vs. doing it yourself. As a programmer, every line of code you write is a line of code you own: the less code you own in production, the better off you are. Algolia has saved us hundreds of hours which translates to tens of thousands of dollars.