If you have a medium amount of data (2GB - 2.4TB), high-security concerns, and search is a key requirement in your single-tenant application then Azure Search likely has you covered. If you have a small amount of data per tenant (EG, about 2GB), have low-security concerns, and a multi-tenant application where search is a key requirement, then Azure Search would likely be a good choice - though you would need to implement your own concept of sharding and managing across potentially multiple Azure Search instances. If you can reflect your would-be indexes in Azure Search by depositing the data in columns in a SQL table and just index it for full-text search - and that still fits your requirements - it's probably better to start with SQL Database then scale up to Azure Search when you need the advanced features like ranking or cognitive abilities.
Elasticsearch is really well suited for searching text (Natural Language Processing) and you can fine tune the searches and scoring very well. I like the ability to find Significant Terms in the Index, where you can find aggregations that are really relevant to a specific search. It also allows for queries to lead to new queries via aggregations which is great for navigating your data. It is less suited to doing more complex aggregations where slices of data are required to be processing using guassian normalizations. And doing searches which join different documents is very very hard, and requires serious thought on how to denormalize data.
Like virtually all Azure services, it has first-class treatment for .Net as the developer platform of choice, but largely ignores other options. While there is a first-party Python SDK, there are only community packages for other languages like Ruby and Node. Might be a game of roulette for those to be kept up-to-date. This might make it a non-starter for some teams that don't want to do the work to integrate with the REST API directly.
In my opinion, partitions inside of Azure Search don't count as data segregation for customers in a multi-tenant app, so any application where you have many customers with high-security concerns, Azure Search is probably a non-starter.
To elaborate on the multi-tenant issue: Azure Search's approach to pricing is pretty steep. While there is a free tier for small applications (50MB of content or less) the first paid tier is about 14x more expensive than the first SQL Database tier that supports full-text search. For many applications, it makes a lot more economic sense to just run some LIKE or CONTAINS queries on columns in a table rather than going with Azure Search.
Setting Java memory thresholds can be a pain for those not accustomed to things like Eden Space & Old Generation which can lead to over allocation, or more likely, under allocation. Apache Solr had a similar issue. It would be nice if the program would take an extra step and dogfood it's own advice by analyzing the system & processes to return a solid recommendation for that configuration. The proper configuration information is outlined in the documentation, it would be nice if that was automated.
The only health check that ElasticSearch reports back is a "red" status without any real solid information about what is going on, though its usually memory thresholds or disk I/O. I am currently on ElasticSearch 1.5 so that may have changed for newer versions. When the status goes "red", I as the administrator of the software, feel like I lose control of whats going on which should rarely happen. Something more verbose would eliminate that.
This is more of a critique of the ElasticStack in general. The whole top to bottom stack is starting to get feature creep with things that are better suited in other software and increasing the barrier for entry for people to get started with setting up a robust logging infrastructure. ElasticSearch as a storage search engine, is pretty streamlined, but I can see that the tools that comprise the ELK Stack are going to require a certification with constant study at some point. During major release for Logstash a while back, it literally took a month to learn a new language because Elastic completely changed the syntax. For a medium sized organization of only a couple of admins, that is a pretty high bar where time is money. They really should work on refining/automating the tools & search engine they have, instead of shoehorning/changing things on to an already rock solid foundation.
To get started with Elasticsearch, you don't have to get very involved in configuring what really is an incredibly complex system under the hood. You simply install the package, run the service, and you're immediately able to begin using it. You don't need to learn any sort of query language to add data to Elasticsearch or perform some basic searching. If you're used to any sort of RESTful API, getting started with Elasticsearch is a breeze. If you've never interacted with a RESTful API directly, the journey may be a little more bumpy. Overall, though, it's incredibly simple to use for what it's doing under the covers.
We've only used it as an opensource tooling. We did not purchase any additional support to roll out the elasticsearch software. When rolling out the application on our platform we've used the documentation which was available online. During our test phases we did not experience any bugs or issues so we did not rely on support at all.
Azure Search is a competitor against Google's own AI autosuggest a feature. We went with Azure because our network security folks found it to be more robust from a security standpoint, which is incredibly important when you have proprietary manufacturing information. Additionally, we're a Microsoft shop so it plugged into our cloud hosting package and client facing OS.
Elasticsearch is the most well-known and supported free data platform that we identified. We are taking advantage of community knowledge and practices. In terms of flexibility and breadth of use cases no other competitor came close to Elasticsearch. We've tried Solr in the past be we encountered issues which were deal-breaking for us. MongoDB - it just did not pass our evaluation parameters as a main data platform. We still use it for smaller purposes, though.
I am not in finance and I suspect even if I was this would be hard to measure. But for sure, Elasticsearch has enabled us to have the most flexible data model in the industry for our customer's data, and in doing so we have attracted many many technical customers and got much of their $$$.
One problem with Elasticsearch is that because it runs on the JVM, there can be some stop-the-world JVM garbage collections happening that can take down nodes and reduce indexing speed. The solution for that tends to be "let's just upgrade the CPU on that machine". And before you know it you are paying $$$ because this'll happen with 40+ machines.
On the other hand, I do think that ES is more efficient than other systems and so it requires fewer nodes to keep it highly tolerant and available, so we probably saved some money that way.