Mongodb fulltext search can do also.
But for better search experience one should go with specific kind of tools/engine made for search only.
I had used ElasticSearch in Practo for some work, now they moved to lucene and other stacks.
Because search specific engines like elasticsearch/solr provide you with advance querying and are optimized highly for search rather than storage. I can say having used Elasticsearch & Mongodb that you can't perform many search queries (which are practically required in search apps, which elasticsearch provides) in mongodb or mysql or any storage specific db.
I have heard Amazon CloudSearch is ElasticSearch of amazon.. like dynamodb is mongodb of amazon.. :) Heard first time about algolia, thank you.