Yes, they can have different meanings. As verb phrases they basically mean the same thing: "She stays home on Monday nights" = "She stays at home on Monday nights". But we can use "stay at home" (or "stay-at-home") as an adjective; for example: "a stay-at-home mom/dad/parent" (a parent who does not work outside the home because they stay home in order to care for their children), or "a stay-at-home job" (some type of work for pay that one can do without leaving one's home).

@Morgan thank you so much:3

They are the same - you can say either! Sometimes parents that do not have a job outside the home will be called a stay-at-home mom/dad, which is the only time I can think of that it makes more sense to say “stay at home” instead of “stay home”. Hope this helps!