"Hack and slash" term originates from pnp RPG culture and was carried over to computer video game culture, primaly to games like Diablo and forthcoming Action-RPGs. Eventually any RPG elements were washed out and a lot of this type of games came from p'n'c, turn-based type of comabt to action-oriented with very shallow RPG elements if any (some basic leveling and limited inventory). While "beat 'em up" was used in regard to arcade fighting games, that includes both versus fighting games and Golden Axe type games initially, but now the term "beat 'em ups" is rarely used to describe versus fighting gameplay.
Today all these games convergent just like animal species in nature due to common conditions though originates from different subsultures (they're basicaly appealing to the same type of gamer), so there're no to little systematic difference in them nowadays. Additionally I believe there were yet another stream of games that contributed to the genre, that originates from console and computer type of action-adventure games (i.e. Zelda-type adventures and early Rare games on C64 and Spectrum respectively) that shifted to action heavily (God of War like games).
Of course, you still can divide these type of games into beat'em up and slasher groups judging by the fact do they use a weapon system of any kind. By weapon system I mean actual difference in using different type of weapons in tactical sense depending on situation with no straightforward route of upgrade and gameplay that emphasize on collecting of these weapons (Monster Hunter type of games, countless MMOs), beat 'em ups even those with weapons usualy don't have any of these gameplay elements.
