Most Empaths are HSP. Not all HSPs (in fact not many) are Empaths.
HSPs are said to make up 20% of the population. Empaths are thought to make up less than 5%.
There is some overlap, but not as much as people seem to think. There are some common traits; but there has been a big confusion online about these two terms and Empath forums are often flooded with HSPs (confusing the two terms and assuming that as they are sensitive and empathic people they are Empaths) telling us more or less to just 'get over it', 'snap out of it', 'be positive', or just recognize we are all one, or similar guff that does not help anyone who is actually an Empath.
A lot of the ridicule and disbelief around such things as distance pickups comes from HSPs who think that whatever they experience is what Empaths experience, and that such things are therefore impossible or rare and only involve blood family in crisis, etc, when for some of us it's an everyday experience; and yet I've also encountered Empaths who refer to themselves as HSPs only, as they find online Empaths whiny special snowflakes and don't identify, and yet who clearly have psychic experiences of the feeling kind, such as Empaths have.