That's an excellent and quite thoughtful question! It gets at the nuances of everyday language. In practical, functional terms, there is no major difference between a "litter bin" and a standard public waste bin. Both are containers placed in public areas for people to dispose of small items of trash, helping to keep streets, parks, and facilities clean.
The distinction is primarily one of regional terminology and slight connotation. "Litter bin" is a term more commonly used in British English and many Commonwealth countries. It directly addresses the problem of "litter" (scattered trash), implying its purpose is to prevent careless disposal. "Public waste bin" or "trash can" (more common in American English) is a more generic, descriptive term for the object itself.
Sometimes, "litter bin" can evoke a specific image of a smaller, often freestanding bin in a park or on a pedestrian street, meant for casual, immediate disposal like wrappers or leaflets. "Public waste bin" might be used more broadly to include larger, fixed receptacles at bus stops, near benches, or with different compartments. However, you will often hear the terms used interchangeably.
So, while you might choose one word over the other depending on where you are, if you're looking for a place to throw away your coffee cup, either one will do the job perfectly. The most important thing is using them, whatever they're called!