Which one to consider for upgrade SQL Server 2008 vs SQL Server 2014 or Compatibility level 100 vs 110
Compatibility levels are the way to tell SQL Server how to interpret the T-SQL and how the database will behave. Paul Randal wrote up a short comparison about the differences between version and compatibility levels about 9 years ago. 

In a nutshell he says there is no relationship between compatibility versions and a database versions.  But you may have access to the new features in an older compatibility version of a database they just may not behave as expected.  One extra thing of note is that if you have multiple databases on the same server you can set each of them to their own compatibility level.