There are reasons in favour of both approaches, with the balance more likely to come out in favour of the 'vanilla' version for may end-users, but in favour of the 'managed' version for server set ups.
In favour of the 'vanilla' TeX Live from TUG, apart from the small 'freeze' period each year, this is updated most days from CTAN. This makes it easy to update your TeX system to get the most recent version of packages, which is often useful to get new features or fix bugs. It also means that you will get most 'free' packages without having to install them yourself. A 'stand alone' installation is managed by
tlmgr, allowing you to selectively add or remove individual packages, etc.
In favour of the 'system managed' installation, this will contain binaries compiled dynamically against system-managed libraries. Thus if an update is necessary to some third-party code, it will automatically apply to the TeX binaries when your general system update takes place. (In comparison, the 'vanilla' TUG version is entirely independent, which means that if there is a bug in some third-part code, TeX Live has to update this and you then have to install the update for TeX Live to fix the issue.) The 'system managed' TeX will also be required by other TeX-dependent packages (e.g. most TeX editors available for Linux), and so installing the managed one is often required/easier if you want to install say TeXworks using your package manager.
A point highlighted in comments, and one of the reasons that for end users a 'vanilla' installation is usually recommended, is the fact that Linux distributions do not necessarily use the latest TeX Live. As such, they will get no fixes for the (La)TeX code they contain (as this does not updated 'across' TeX Live versions). They do get the library fixes I've mentioned, which are important for security, but for most end users it's changes in (La)TeX packages that are significant. Even if a distro is using the latest TeX Live version, the need to repackage material almost certainly means that updates to (La)TeX material will lag behind those in 'vanilla' TeX Live.
The question asks which set up is more 'stable'. That depends on what you want. If you never update installed (La)TeX packages, and only change the binaries for security reasons, then 'vanilla' TeX Live is essentially identical in this respect to a managed installation. However, being able to add/update installed packages does in that sense reduce stability (change packages => change output).