I also wanted to understand the difference between להניח and לשים . The website Morfix says להניח can also mean 'to deposit' or 'to set down'. Those definitions might be a better way of thinking about the difference.
In English you can also use "to rest" as an intransitive (i think??) verb, similar to "to put" or "to place." Something like, "He rests his feet upon the table." This is probably a related use of the root נח ?