Do I really Need to Upgrade to SQL Server 2014 SP2 ? -
Your manager will be concerned about possible regressions and new bugs in the service pack (rare, but it has happened in the past) or one of your applications relying on an undefined behaviour that is changed by SP2. These are valid concerns and it is not recommended that you upgrade in production without first at least reading all the official change documentation to make sure none of your applications are likely to be affected. Fully testing your application(s) against the upgrade in another environment is strongly recommended too.

If you have third party applications you might not have the product knowledge to perform the above checks/test, in which case you will need to discus the matter with the application vendor. Hopefully they can give you definitive word on whether they support their application(s) when used with SQL Server 2014sp2.

Do note that without SP2 SQL 2014 becomes officially unsupported in October according to the official lifecycle (https://support.microsoft.com/en-gb/lifecycle/search?alpha=sql%202014) so you are going to want to install SP2 (or upgrade further) before then, meaning you can't avoid this effort completely.

The chance of significant problems between service packs is much lower than between full releases so many people fly by the seat of their pants and just upgrade to the latest service pack without testing. This attitude is somewhat too risky IMO.