According to the Chicago Manual of Style, “names of secular and religious holidays or specially designated days or seasons are capitalized.”
Easy enough. That includes the day for mothers and also the day for fathers—both “specially designated days.”
But what about that darn apostrophe?
I guess it depends on if you’re talking about a day – in the possessive sense – that belongs to just one mother (mother’s) or to all mothers (mothers’). Or how about just making it a plural day for all mothers, without any possession by any of the mothers: Mothers Day.
For what it’s worth, the Chicago Manual says “Mother’s Day.” My personal preference is "Mothers Day" -- a day that celebrates lot of mums, but that does not belong to anybody.
And the same goes for dads.