The installer does all the work automatically. For a simple non-customised Jira install, the installer can do everything except click "update" on all the apps you have installed.
The manual upgrade needs you to do all the work of installing from a pile of files, so there's a load of files to edit and get right. But it does get you to the same place as the installer in the end.
In both cases, they destroy all your customisations done at the file level, so you will have to re-apply all of them back before letting Jira start up. This is stuff like SSO config, the stuff you need to do to run Jira behind a proxy or load-balancer, config tweaks in the jira properties files, any tweaks you've made to velocity templates and so on.
The installer has another advantage in that it will list all the files you may need to re-apply changes to before it goes on to the next step.