When you'd like to create movement and variation but find an LFO too persistent and predictable, the Brownian LFO is there to offer control voltages that wander as a particle in a glass of water, generally meandering back and forth with idiosyncratic aimlessness along the way. Unlike noise, where each output voltage is independent of the last, Brownian motion is always derived from previous values, creating a more natural, organic change that still offers randomness.
It can be used to add a little bit of random variation for subtle analog-like behaviors (especially with pitch tracking and filter cutoff), large-scale evolutions in tone, or quick and dramatic effects alike. The boundary controls allow you to specify the upper & lower limits of the output voltage, the size control determines how much the voltage can swing in a given moment, the time control sets how often directional changes can occur, and the glide dial offers smoothing of the output. A trigger to the "recenter" jack pushes the output voltage back toward zero to allow for moments when you need to "unwander".
Use the Brownian LFO as a source of control voltages that don't see quite so repetitious, to use controlled randomness for nonstatic sound, and inject a little bit of surprise into your patches.