As with curaçao, the only difference between clear and colored varieties is the color. You can use them interchangeably, as long as you don't mind the possibility of your drink looking funny.
But why are they called crème liqueurs? The white version of both isn't white per se, but crystal clear. Obviously, dairy products haven't come close to either bottle. It's because of the texture of the liqueur. Crème liqueurs have plenty of sugar in them, enough to give them an almost creamy consistency. Many spirits makers also add a touch of glycerin to give the liqueurs some extra body and creaminess.
To add to the confusion, many makers of chocolate liqueurs also make a chocolate cream liqueur, which contains actual dairy products. The quick and dirty way to tell the difference is in the spelling: Crème has no dairy, while cream has dairy.
A well-stocked commercial bar should have both varieties of each spirit. The only reason to have both white and green crème de menthe (or white and brown crème de cacao) in a home bar is to show off. You can certainly get by with just one bottle of each at home. But which ones?