Drush, Brush and Bee are all command line tools for Backdrop CMS. Use of command line tools is not required for Backdrop CMS, however for those are comfortable working on the command line and/or work frequently on Backdrop (support many Backdrop sites) the use of command line tools can be very helpful. Some regular developers might consider them essential.
Here is quick explanation of each (as I understand it).
Drush - The Drush version of Backdrop is a direct extension of Drush for Drupal. It supports a variety of commands that any Drupal 7 user would be familiar with, but only a limited number. At some point, development of Drush for Backdrop shifted towards what is now called Bee. Bee is intended to be a new CLI tool for Backdrop that was written specifically for Backdrop and not inherited from Drupal.
Bee - The CLI tool previously known as "B" has been renamed "Bee" and is intended to replace Drush as the primary CLI tool for Backdrop. It has been rewritten from the ground up to focus specifically on Backdrop. Development of Bee has progressed in periodic bursts and a number of commands work well on it today. However, it doesn't have all the features that Drush did for Drupal 7.
Brush - Brush is being maintained by a group of developers that wanted a number of Drupal 7 features in Drush quickly, but didn't want to wait for Bee to mature. They basically forked Drush for Backdrop and quickly added the features that they most needed. According to maintainer, Alan Mels, the intention was to use Brush as a bridge until Bee matured. See this forum topic for a more detailed history of Brush and discussion of it's future.
Personally, I'm not quire sure how far Bee has matured, but would welcome someone giving us a more up to date comparison of Bee and Brush in terms of features.
I have experimented and used both Drush for Backdrop and Bee. Neither have the sync features that I so very much appreciated in Drush for Drupal. I understand that these sync features are available in Brush and I may finally install it for the first time. It's not clear to me if "Bee" plans to support these same features.
Having said all of that, I still plan for Bee to become the default CLI tool for Backdrop, but in the meantime I continue to use Drush for Backdrop and may start using Brush very soon.
We recorded a Virtual User Group on this topic in March of 2021.