I'd say Hard is only noticeable when you do the boss battles. Nightmare, on the other hand, is noticeable for some mobs and most rifts, as well as bosses.
Hard and especally nightmare will require you to take certain critical abilities to make a run easier, or at least it is highly recomended you do, like warrior's horn of valor with fortifying blast upgrade (on your warrior), 2 mages both with barrier spells and a rogue with the toxic cloud ability and its elusive stacks upgrade. With those abilities and upgrades (and setting Ai to use them) you are set. Combine it with crafting (make sure you craft weapons and armor with best tier 3/ tier 4 materials) and you are set. That is all you really need. Seriously, people can solo this game on nm+trials so a full party of 4 toons will get the job done. Besides if one of the mages has the revive spell you can always revive fallen companions. And if that does not work just "retreat" with the last surviving character from a difficult encounter and come back when you are more leveled.
Adding to that excellent advice are a few more tips recorded here:
(GUIDE) Simple tactics for new players -- how to survive until SkyholdYou're not a new player, but the tips will still be helpful for a Hard/Nightmare run. Make sure you read the whole thread, since there are some alternatives and further discussion of better tips in replies.
In that guide is a link to another guide, Making Short Work of the Hinterlands, that will speed things up for you. Particularly if you are going to do a Hard and a Nightmare run.