Search API vs. Apache Solr Search
I've tried using both and I can say this: it depends on your situation.

Currently, the stable 7 release of the ApacheSolr Integration module can only index nodes.  So if you have non-node entities that you need to index, you have to use the still in progress multientity patch for it.  ApacheSolr Integration can store a lot of different data of content when configured properly.

The Search API does index entites and has a lot of wonderful stuff written for it.  However, Search API only fetches the id of the data you are searching for.  This means to load any more data other than the ID will require an entity_load, hitting your database or whatever caching layer you put in place.  For search-heavy sites, this might not be the most optimized solution.

Here is a great presentation given at drupalcon chicago about the ApacheSolr Integration module, minute 16 for mentions to Search API.