SQL Maintenance Plan runs successfully but files not backed up to folder

Check the maintenance plan log and see what's in there
Using SSMS, double click on your Maintenance Plan in the Object Explorer. In the Object Explorer Details, click on the "Reporting and Loging" icon (close to the "Manage Connections" button). Go to that path, and find the log file

Any errors or otherwise helpful info in the error log for the sql agent job associated with the maintenance plan?

Checking the reporting and logging I do see the error message

"2013-09-20 11:07:10 - ! [382] Logon to server 'E2PBLD01' failed (ConnAttemptCachableOp)"

How is that possible even though I am connected?

Check the connection settings on your maintenance plan.

Can you post a snapshot of the connection that the SSIS package is using as well as check what service account IS and/or SQL Server Agent are running as? (when you open the maintenance plan, it will be in Manage Connections...)  If it's trying to access a remote machine and the integration services account is installed as a local system account that would present an issue and also check the effective permissions of the SQL Server Agent's service account if it's a local connection.

When going back to check the maintenance plan I now get the message

"Microsoft SQL Server Management Studio is unable to load this document.  The LoadFromSQLServer method has encountered OLE DB error code 0x80004005"

I have reinstalled client tools and integration services.  Is it possible something is inherently wrong with the SQL installation?

login as another windows user and try to load the plan

It allows me to run it successfully.  However it only takes 2 seconds to run and checking the maintenance plan log I get the same error:

"2013-09-20 15:16:10 - ! [382] Logon to server 'E2PBLD01' failed (ConnAttemptCachableOp)"

Any ideas?

Open the maintenance plan, make a new connection, manually enter login info and save it. then run it