Chris Sommer points out that the SQL Browser does something useful:
Bingo. The application server could not connect to the SQL Browser service on UDP 1434. So maybe now you’re asking why, and that’s kinda the gist of this post. The SQL Browser provides a valuable service when an application tries to connect to a SQL Server named instance. The SQL Browser listens on UDP 1434 and provides information about all SQL Server instances that are installed on the server. One of those pieces of information is the TCP port number that SQL is listening on. Without that info, the application has no idea how to reach to your SQL Server, and will fail to connect. This was our exact issue.
Do read this, though my preference is to shut off the SQL Browser because it’s a mechanism attackers can use for gathering intel on where SQL Server instances live.