Basically, the SUID bit makes a program get the permissions of the owner when executed. If you set /bin/bash as SUID, suddenly every bash shell would be a root shell, kind of. Processes on Linux have a real user ID, an effective user ID, and also a saved user ID that can be used to temporarily drop privileges and gain them back again later.

So tools like sudo and doas use this mechanism to temporarily become root, then run checks to make sure you're allowed to use sudo, then run your command. But that process is still in your user's session and process group, and you're still its real user ID. If anything goes wrong between sudo being root and checking permissions, that can lead to a root shell when you weren't supposed to, and you have a root exploit. Sudo is entirely responsible for cleaning the environment before launching the child process so that it's safe.

Run0/systemd-run acts more like an API client. The client, running as your user, asks systemd to create a process and give you its inputs and outputs, which then creates it on your behalf on a clean process tree completely separate from your user session's process tree and group. The client never ever gets permissions, never has to check for the permissions, it's systemd that does over D-Bus through PolKit which are both isolated and unprivileged services. So there's no dangerous code running anywhere to exploit to gain privileges. And it makes run0 very non-special and boring in the process, it really does practically nothing. Want to make your own in Python? You can, safely and quite easily. Any app can easily integrate sudo functionnality fairly safely, and it'll even trigger the DE's elevated permission prompt, which is a separate process so you can grant sudo access to an app without it being able to know about your password.

Run0 takes care of interpreting what you want to do, D-Bus passes the message around, PolKit adds its stamp of approval to it, systemd takes care of spawning of the process and only the spawning of the process. Every bit does its job in isolation from the others so it's hard to exploit.

SUID stands for Set User ID. An SUID binary is a file that is always run with the UID of the owner user (almost always root). Note that this does not require that the user running them has root permissions, the UID is always changed. For instance, the ping command needs to set up network sockets, which requires root permissions, but is also often used by non-root users to check their network connections. Instead of having to sudo ping, any normal user is able to just run ping, as it uses SUID to run as the root user. sudo and doas also require functions that necessitate them running as root, and so if you can find out how to exploit these commands to run some arbitrary code without having to authenticate (since authentication happens after the binary has started running), there is a potential for vulnerabilities. Specifically, there is the privilege escalation, which is one of the most severe types of vulnerabilities.

run0 starts using systemd-run, which does not use SUID. Instead, it runs with the permissions of the current user, and then authenticates to the root user after the binary has already started to run. systemd-run contacts polkit for authentication, and if it succeeds, it creates a root PTY (pseudo-terminal/virtual terminal), and sends information between your session and the root PTY. So this means that in order to achieve privilege escalation with run0 as root, you have to actually authenticate first, removing the "before authentication" attack surface of sudo and doas.

TL;DR SUID binaries will always run as the owner (usually root), even before any form of authentication. run0 will start with the permissions of the current user, and then authenticate before running anything with root permissions.

Suid is a bit set on executables that results in them being run as the user that owns the file without needing a password, for example, passwd as root.

Run0 ignores this bit