Fear is an attack on the Morale of another being. It has many sources, of which the Fear Aura is the most common. The core purpose of Fear is force a rout in the enemy, greatly reducing the amount of bodies one has to cleave through in their bid for domination or survival.
Units with THIS icon in their stat sheet have a Fear Aura. Each round, a unit with an Aura has just under 24 chances to cover either the square they're on or an adjacent square (including diagonals) with its Aura. The listed value of a Fear Aura, plus 6, is a little under how many times a Fear Aura will flash a random square each round.
How much of an effect does a Fear Aura have? Well…
In Dominions 6, Fear operates differently. It has been made 3 points weaker, lowering Morale by less (zero at Fear +0), and it can no longer lower Morale more than 10 points below its base value.
On the other hand, Fear Auras always cause a Morale roll for a rout now; the difference is that the difficulty of the check is significantly lower. The base score for the so-called "Fear roll" is a 10 if it's caused by a Fear +0 Aura. Every 5 points added to the aura increases the base by 1; a Fear +5 Aura (from a really-intimidating human or giant) has a base of 11, for instance, while a Fear +10 Aura (from a Dragon or something similar) has a base of 12. It's not clear if this caps at the traditional base of 14, which Fear +20 would inflict.
On top of that, the unit(s) directly "flashed" by a Fear Aura may individually rout, if their personal Morale fails an "easy" roll. "Easy" is revealed by game-testing to be a 6, regardless of the strength of the Fear Aura; while this would normally be very easy to circumvent, the check is taken after the Morale loss. There is a small delay between losing Morale and taking the rout check, however, in which some lucky units may have their Morale correct slightly. An individual routing does not cause their entire squad to rout, though it may cause issues if it brings the squad down to four members (or below other thresholds).
Dread is new to Dominions 6. It's equivalent to Fear in almost all respects, except that units with True Sight (or Spirit Sight, or Blindness) aren't affected by it. If a unit has both, only whichever has the higher value (if it applies to the target) is used.
Why use Dread? Quite simply, Dread is far easier to apply than Fear. It's a Glamour effect that comes in two varieties. There's the Bless variety, which provides Dread +5, and then there's the spell variety that Glamour/Death Mages can apply. According to the Mod Inspector, the two Dread-applying Enchantments only grant Dread +0, but the "mechanical" yellow text on the spells says they apply Dread +5. Compare this to Fear, which can only be added to units by items and a Death Bless.
Spells and certain magic weapon attacks can cause Fear in bursts, directly attacking the Morale score and possibly causing a rout. This is sometimes called "damage", though it does not reduce Hit Points. These come in two types:
Fear is a not-well-liked part of Dominions 6. Most attribute the distaste to it being much more common than before, thanks to the Bless being made cheaper, and to Dread being a thing. Perhaps the massive increase in the number of Morale rolls is too much for the game, despite the 2-to-3-point decrease in its effectiveness. Maybe its impact just decays too slowly. It could just be a shift in the metagame that the old Dominions 5 players aren't used to; watching one's multi-buffed super-soldiers get turned around after a few rounds and scratches can certainly leave a bad taste in the mouth.
Fear does not work on the Mindless or the Berserk, and a great many Demons have the maximum amount of standard Morale (30) for resisting such things. It is not clear if 30-Morale beings can be routed by Fear at all, as long as their squad consists solely of 30-Morale beings.
Aside from that, one can try their best to keep average squad Morale high, by deploying bigger squads or taking the Inspirational Presence Bless. In battle, penalties to Morale can be burned away with Divine Magic, with more 2 priests getting the job done faster.
Alternatively, a fellow relying on Fear could be countered through a thorough nuking with Evocations. Taking Fear or Dread on the Bless comes at the cost of points that could go into Magic Resistance or Elemental Resistances, after all. The paths that bestow Dread don't lend themselves to many defenses against magic.