We are developing an in-house 32bit app. The app connects to a SQL Server. The test environment is SQL Server 2008 R2, and the live environment is SQL Server 2014 SP2.
Thanks to Birel for pointing me in the right direction. The answer was that on the new 2014 server, the 'SQL Server Browser Service' was not running. I set it to automatic startup, and started it. That resolved the issue.