Yes. You can have MOVES2010b and MOVES2014 on the same computer at the same time. Running MOVES2010b using the updated MySQL 6.5 database application will not affect the results you have been getting with MOVES2010b and you can have both default databases in your set of database files.
The issue with MOVES2010b is that the older database application (MySQL 5.5) was not configured to provide adequate security to allow it to be installed on current networks, given the latest security requirements. When you install MOVES2014 (and MySQL 5.6) a new level of security will be required, including a password for access to MySQL. The “patch” to MOVES2010b allows it to provide the passwords needed to use MySQL in the new setup. Nothing else about the operation or results of the model will be affected.