hey mate stay does not imply a choice
stay silent is "imperative" and so is "remain silent"
remain just sounds a bit more formal.
if you were reading a fire escape play it might say "remain calm" but "stay calm" would also be fine
once again.. stay does not imply choice
you can also say "wanna stay at my house tonight?" but you can't use remain here
also "it remains a mystery" instead of using "stay"