A restroom (American English) is basically a scaled down bathroom, consisting of a toilet, sink, and a mirror. A home may have to separate toiletry function from the bathroom, or in home designed on the public and family area separation, so that guests can use it without needing to intrude on the family space.
Public restrooms tend to scaled up versions on the private versions, but to handle multiple people rather than functions of a bathroom. Many cultures insist on public restrooms being gender separated. Male restrooms will have few toilet stalls and a large or multiple urinals, whereas female restrooms will have more stalls, and no urinals, but perhaps a couch instead. Traditionally the only female restrooms would have have a baby changing station, but it is becoming more common for restrooms for both genders to have such stations.
Even in cultures that insist on gender separation there will me the occasional uni-sex restrooms designed to cater the needs of younger children, these can be referred to as family rooms (not to be confused with the room in the home with the same name).