Yes, these are two different types of integrations. The Add integration is a basic integration and only opens the add-issue form. This is a generic integration and supports almost all web-based bug and issue tracking tools. The Push integration on the other hand is much more advanced and allows you to add new issues directly from TestRail’s user interface. TestRail currently supports this integration for the ~15 most popular issue and bug tracking tools (JIRA, Redmine, Rally, Axosoft, FogBugz, GitHub, Bugzilla and many more).
Since you use JIRA we can recommend using the Push and Lookup integration. The integration is very flexible and customizable and supports custom fields on the Push dialog as well as different configurations per TestRail project etc.
I hope this helps!