“Working from home” is the phrase typically given to one of two things. The first is a temporary action of working in your home, rather than the office, maybe due to extraordinary circumstances like a hurricane or snowstorm.
As an example, I feel like you’d say, “The roads aren’t plowed, so I’m working from home today.”
Second, working from home can mean working remotely as a more permanent situation, but specifically from one’s place of residence. Some people are the single (or one of few) remote employees in a larger company that has tangible office addresses. They work from home because their skills are needed, and they happen to not be located within a commutable distance from said office.
Working remotely is (to me) more of a permanent circumstance. Some people work remotely by choice, and conversely, some don’t.
Working remotely seems to be something millennials want because it represents freedom – freedom to work from somewhere that specifically is not home, nor an office. While this of course applies to non-millennials too, it’s common to say you’re “working remotely” if it’s from a cafe, a place you’re traveling or living abroad, or lastly, a coworking space.
Working remotely, or being a remote worker, can also become a mindset of sorts. If you’ve been out of a traditional office long enough, you start standing by the ideas that “work doesn’t have to take place in an office” and that time spent behind a desk next to a colleague doesn’t necessarily mean you were doing work the whole time anyway (yikes).