Flyweight is all about improving performance and memory usage. All embedded systems (phones, tablets, game consoles, microcontrollers, and so forth) and performance-critical applications (games, 3D graphics processing, real-time systems, and so forth) can benefit from it.
The Gang Of Four (GoF) book lists the following requirements that need to be satisfied to effectively use the Flyweight Pattern [GOF95, page 221]:
The application needs to use a large number of objects.
There are so many objects that it's too expensive to store/render them. Once the mutable state is removed (because if it is required, it should be passed explicitly to Flyweight by the client code), many groups of distinct objects can be replaced by relatively few shared objects.
Object identity is not important for the application. We cannot rely on object identity because object sharing causes identity comparisons to fail (objects that appear different to the client code, end up having the same identity).