Question

What are the core architectural differences between these technologies?

Also, what use cases are generally more appropriate for each?

Answer 1

Update

Now that the question scope has been corrected, I might add something in this regard as well:

There are many comparisons between Apache Solr and Elasticsearch available, so I’ll reference those I found most useful myself, i.e. covering the most important aspects:

• Bob Yoplait already linked kimchy’s answer to ElasticSearch, Sphinx, Lucene, Solr, Xapian. Which fits for which usage?, which summarizes the reasons why he went ahead and created ElasticSearch, which in his opinion provides a much superior distributed model and ease of use in comparison to Solr.

• Ryan Sonnek’s Realtime Search: Solr vs Elasticsearch provides an insightful analysis/comparison and explains why he switched from Solr to ElasticSeach, despite being a happy Solr user already – he summarizes this as follows:

Solr may be the weapon of choice when building standard search
applications, but Elasticsearch takes it to the next level with an
architecture for creating modern realtime search applications.
Percolation is an exciting and innovative feature that singlehandedly
blows Solr right out of the water. Elasticsearch is scalable, speedy
and a dream to integrate with. Adios Solr, it was nice knowing you. [emphasis mine]

• At the time this answer was written (2012), the Wikipedia article on Elasticsearch summarized a comparison from the reputed German iX magazine, listing advantages and disadvantages, which pretty much summarizes what has been said in other answers already:

Advantages:

• Elasticsearch is distributed. No separate project required. Replicas are near real-time too, which is called "Push replication".
• ElasticSearch fully supports the near real-time search of Apache
  Lucene.
• Handling multitenancy is not a special configuration, where
  with Solr a more advanced setup is necessary.
• ElasticSearch introduces
  the concept of the Gateway, which makes full backups easier.

Disadvantages:

• Only one main developer [not applicable anymore according to the current elasticsearch GitHub organization, besides having a pretty active committer base in the first place]
• No autowarming feature [not applicable anymore according to the new Index Warmup API]

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Initial Answer

They are completely different technologies addressing completely different use cases, thus cannot be compared at all in any meaningful way:

• Apache Solr – Apache Solr offers Lucene’s capabilities in an easy to use, fast search server with additional features like faceting, scalability and much more

• Amazon ElastiCache – Amazon ElastiCache is a web service that makes it easy to deploy, operate, and scale an in-memory cache in the cloud.

• Please note that Amazon ElastiCache is protocol-compliant with Memcached, a widely adopted memory object caching system, so code, applications, and popular tools that you use today with existing Memcached environments will work seamlessly with the service (see Memcached for details).

[emphasis mine]

Maybe this has been confused with the following two related technologies one way or another:

• Elasticsearch – It is an Open Source (Apache 2), Distributed, RESTful, Search Engine built on top of Apache Lucene.

• Amazon CloudSearch – Amazon CloudSearch is a fully-managed search service in the cloud that allows customers to easily integrate fast and highly scalable search functionality into their applications.

The Solr and Elasticsearch offerings sound strikingly similar at first sight, and both use the same backend search engine, namely Apache Lucene.

While Solr is older, quite versatile and mature and widely used accordingly, ElasticSearch has been developed specifically to address Solr shortcomings with scalability requirements in modern cloud environments, which are hard(er) to address with Solr.

As such it would probably be most useful to compare ElasticSearch with the recently introduced Amazon CloudSearch (see the introductory post Start Searching in One Hour for Less Than $100 / Month), because both claim to cover the same use cases in principle.