What do you like best?
TitanDB uses Gremlin 3, the new version of a good query language with tons of documentation, slides and videos on youtube. You can store your data in Cassandra, which means using a very well known database without special nodes (single point of failure). You can do it in Hbase on a Hadoop cluster (compatibility with Hadoop v1 and v2), very convenient for very large datasets or in BerkeleyDB.
What do you dislike?
TitanDB is quite complex at the beginning and it can be challenging for not very experience people as you need to deploy not only the Titan server but also a database and a search engine (Solr, ElasticSearch and Lucene). This can be challenging specially when you try to automate deployment or installation. Also, TitanDB is 1.0.0 which means a very young project prone to bugs and that should be in production with care in the beginning stages.
Recommendations to others considering the product
TitanDB is only for experience users and companies that really know what they are doing or for users that have such a huge database that cannot be easily stored in Neo4J. It's nice documented and has some examples on the webpage but it's definitely not a database to newcomers to the graph world.
What business problems are you solving with the product? What benefits have you realized?
TitanDB solves data analysis on very large datasets stored on HDFS or in datasets already stored in Apache Cassandra that can be queried without big problems all will take advantage of a commodity cluster capabilities to distribute workload between all machines.