Must have is something that many people would want, almost a necessity (that is, something that is (almost) necessary or indispensable. ) But it depends, sort of, on the judgement of the speaker, since what is a must-have to someone may not be to someone else. Things that you must own/that you should have with you, in your wardrobe/closet, in your kitchen, in your car etc. Since they leave your life easier.
An example: "a coffee machine is a must-have in the office environs." (in fact, it may or may not, if you think of it.)
Tongs are a must-have around the kitchen, they are such a workhorses! Tongs can be used for picking up salad, turning bacon, lifting hot items and serving.
(it emphasizes the time you save by having them in your kitchen; if you don´t have them, you´ll have to pick up that ice with a spoon, or to pick that macaroni with other utensil (say with a fork or two, for example.) http://www.yourdictionary.com/must-have ... anheritage
Must have (without hyphen) discussion.http://forum.thefreedictionary.com/post ... ence-.aspx
Mandatory would be something that is required, necessary, obligatory. It is "something that has to be done" (as per Cambridge´s Dic. definition), so rather stronger idea than "must-have".
The meaning of "mandatory" being more of "obligatory or essential to something", and "a must-have" being more of "something strongly advised to be done/bought/purchased, etc."