I agree with @IMSoP's assertions above...
To compare the two is like comparing an elephant and a tiger - you're not really asking the right question quite yet.
What you should really be asking is - what are my requirements for my use cases to best fulfill my stakeholder / customer needs, first, and then which data storage technology best aligns with my requirements second...
To be clear - Whether speaking of AWS ElasticSearch Service, or FOSS / Enterprise ElasticSearch (which have signifficant differences, between, even) - ElasticSearch is NOT a Relational Database (RDBMS), nor is it quite a NoSQL (Document Store) Database, either...
ElasticSearch is a Search Engine / Index. It does some things very well, for very specific use cases, however unlike RDBMS data models most signifficantly, ElasticSearch or NoSQL are not going to provide you with FULL ACID Compliance, or Transactional Statement Processing, so if your use case prioritizes data integrity, constrainability, reliability, audit ability, regulatory compliance, recover ability (to Point in Time, even), and normalization of data model for performance and least repetition of data while providing deep cardinality and enforcing model constraints for optimal integrity, "NoSQL and Elastic are not the Droids you're looking for..." and you should be implementing a RDBMS solution. As already mentioned, the AWS Redshift Service is based on PostgreSQL - which is one of the most popular OpenSource RDBMS flavors out there, just offered by AWS as a fully managed solution / service for their customers.
Elastic falls between RDBMS and NoSQL categories, as it is a Search Engine / Index that works most optimally with "single index" type use cases, where A LOT of content is indexed all at once and those documents aren't updated very frequently after the initial bulk indexing,but perhaps the most important thing I could stress is that in my experience it typically does not scale very cost effectively (even managed cluster services) if you want your clusters to perform well, not degrade over time, retain large historical datasets, and remain highly available for your consumers - and for most will likely become cost PROHIBITIVE VERY fast. That said, Elastic Search DOES still have very optimal use cases, so is always worth evaluating against your unique requirements - just keep scalability and cost in mind while doing so.
Lastly let's call NoSQL what it is, a Document Store that stores collections of documents (most often in JSON format) and while they also do indexing, offer some semblance of an Authentication and Authorization model, provide CRUD operability (or even SQL support nowadays, which makes the career Enterprise Data Engineer in me giggle, that SQL is now the preferred means of querying data from their NoSQL instances! :D )- Still NOT a traditional database, likely won't provide you with much control over your data's integrity - BUT that is precisely what "NoSQL" Document Stores were designed to work best for - UNSTRUCTURED DATA - where you may not always know what your data model is going to look like from the start, or your use case prioritizes data model flexibility over enforcing data integrity in general (non mission critical data). Last - while most modern NoSQL Document Stores may have SOME features that appear on the surface to resemble RDBMS, I am not aware of ANY in that category at current that could claim to offer all that a relational database does, with Oracle MySQL's DocumentStore being probably the best of both worlds in my opinion (and not just because I've worked with it every day for the last decade, either...).
So - I hope Developers with similar questions come across this thread, and after reading are much better informed to make the most optimal design decisions for their use cases - because if we're all being honest with ourselves - everything we do in our profession is about data - either generating it, transporting it, rendering it, transforming it....it all starts and ends with data, and making the most optimal data storage decisions for your applications will literally define the rest of your project!
Cheers!
