RTIC is a real-time framework, it bridges the gap between bare-metal and RTOS. Because RTIC uses hardware interrupts, it is the most efficient real-time solution with the smallest footprint. The Bern RTOS in contrast, is similar to a classical RTOS by using a preemptive time-sliced scheduler. As the Bern RTOS will have more control over a system than RTIC, it should be possible, for example, to manage power states without user interaction.
Drone OS is probably comes closest to classical RTOS like μC/OS-III that exist in Rust. Unlike the Bern RTOS, which tries to use the existing embedded-hal, Drone OS has a custom register model based on CMSIS-SVD files.
Tock OS uses a micro kernel and is thus the most secure and stable system. This leads to many system calls which worsen real-time performance, but the system was not designed for that purpose. Tock OS, similar to Zephyr, builds hardware support into the system. The Bern RTOS is also set out to be fail-safe, however, some safety features (some drivers will run in kernel mode) are traded for real-time performance.
In summary the Bern RTOS draft fits somewhere between Drone OS and Tock OS.