what does apt update actually update?

Apt update simply refreshes the package list. When packages update their URL for download can change, and the system won't know if anything needs updating without running an apt update. It's more accurate to call apt update a "refresh", since it's a refresh of the package list.
apt upgrade is what actually updates packages on the system, like steam.
When applications upgrade, generally if the application is already running everything keeps working, but you need to restart it to fully upgrade. However there are exceptions for this.
If the application uses shared libraries and the library had breaking changes, it can cause the application to crash or behave in weird ways.
The debian and ubuntu way of doing upgrades is to not have breaking changes for an upgrade. So libraries shouldn't actually change compatibility for example. This is what the meaning of "stable" is with a "stable" distro like ubuntu vs a rolling distro.
In rolling distros, like debian sid or arch, sometimes the entire system can become unusable depending on what has upgraded since libraries and dependencies can change, breaking functionality, sometimes even impossible to shut down properly . Unless there are bugs though, restarting everything fixes it.