I'm not super-familiar with what MongoDB has to offer (I have to catch up on that), but Elasticsearch's strongpoints are related to real-time search and analytics. Some example use-cases that fit Elasticsearch very well:
- log centralization. This is by far the most popular. You can drop lots of logs in there (or any other time-based data, really, could be metrics, for instance) and "grep" through them very quickly, and you can also do lots of statistics. If you want an example (this is actually a product I've been working on) go to https://apps.sematext.com/demo
to the Logsene tab. Logsene is a logging SaaS with an Elasticsearch backend. You can click on the Kibana button there to explore logs through the open-source Kibana UI, which was built specifically for slicing and dicing logs stored in Elasticsearch. If you also heard about Logstash (that can mangle events on their way to Elasticsearch), together they make up what is called the "ELK stack", which is used by many for centralizing logs. There are lots of other tools in the logging ecosystem that work with Elasticsearch. rsyslog is one of them, I'm a big rsyslog fan
- social media. This is another kind of time-based data, but I'm putting it separately because you might have other search needs. Like stemming or fuzzy searches (or statistics - for example you may not want to count "search" and "searching" separately)
- product search. This is more towards the typical search engine case: you have a bunch of products, how do you build relevant search on top of that? Elasticsearch has tons of features in the way text is analyzed (i.e. make tokens from the original text and from the query string and match them) and the way you can run queries. For example, you can rank more exact matches higher, or you can rank newer or promoted products higher
Depending on your use-case, Elasticsearch may have advantages in the search/analytics performance area (because of the way it indexes everything) or in the search relevancy area. That's how I'd divide them, at least.