I am trying to decide whether I should use App-engine Search API or Datastore for an App-engine Connected Android Project. The only distinction that the google documentation
The datastore is a transactional system, which is important in many use cases. The search API is not. For example, you can't put and delete and document in a search index in a single transaction.
The datastore has a lot in common with a NoSql DB like Cassandra, while the search API is really a textual search engine, very similar to something like Lucene. If you understand how a reverse index works, you'll get a better understanding of how the search API works.
A very good reason to combine usage of the datastore API and the search API is that the datastore makes it very difficult to do some types of queries (e.g. free text queries, geospatial queries) that the search API handles very easily. Thus, you could store your main entities in the datastore, but then use the search API if you need to search in ways the datastore doesn't allow. Down the road, I think it would be great if the datastore and search API were more tightly integrated, for example by letting you do free text search against indexed Text fields, where app engine would automatically create a search Document Index behind the scenes for you.