The main advantage Solr has over ElasticSearch
In one word: community
Solr has an extremely vibrant and robust community, and this should be a major consideration in deciding which product to base your application on. Solr has a number of very active committers, and the likelihood of Solr development falling away in the near future is slim.
At the time of writing, ElasticSearch has one active committer (Shay Banon). Although he appears to be very active in fixing bugs and implementing new features, ElasticSearch's fate is very closely tied to Shay's, and this does ask questions as to the project's longevity.
Another area where Solr appears to be further along is customizablity
Whilst ElasticSearch has a number of plugin-points such as modules, index modules, rivers etc, there doesn't appear to be an equivalent of Solr's SearchComponent (see this google groups thread).
If you're doing anything more than just searching for keywords, this could potentially be a deal-breaker in deciding to go with ElasticSearch. SearchComponent provides the developer astonishing flexibility in the way search queries are assembled and executed. At the time of writing, there does not appear to be a ElasticSearch equivalent of SearchComponent.