Numerous spells, abilities, and actions change
The boss is stronger in strange ways, because they...
- Can try to save four times per round against hold monster and similar save-or-suck spells. Other effects, like a monk's Stunning Strike, are unaffected.
- Can grapple or shove four times per round. (One cannot substitute a grapple for an attack within Multiattack, unlike Extra Attack.)
- Are less affected by one bad initiative roll. A good initiative roll isn't nearly as important as avoiding being the last creature to act.
- Can stand up from prone four times per round.
- Negate surprise after just one of their four turns in the first round.
- Have four chances to beat the Assassin rogue, thus eliminating the rogue's advantage.
- Can Disengage exactly when they need to, while only using one of their four attacks.
- Can cast four spells if they have spellcasting ability. Typically, casting a one action spell takes the place of all of a creatures attacks.
The boss is weaker in strange ways, because they...
- Are forced to save four times per round or take damage from spirit guardians and similar damage spells.
- Are much more quickly affected by save-X-times-or-suck spells, like contagion and flesh to stone.
Legendary Actions are the normal answer
As the other answers stated, Legendary Actions avoid all of these problems by allowing a Legendary creature to act outside of its own turn. An Adult Red Dragon is one such Legendary creature and has the following Legendary Actions:
Legendary Actions
Can take 3 Legendary Actions, choosing from the options below. Only one legendary action can be used at a time, and only at the end of another creature's turn. Spent legendary actions are regained at the start of each turn.
Detect: The dragon makes a Wisdom (Perception) check.
Tail Attack: The dragon makes a tail Attack.
Wing Attack (Costs 2 Actions): The dragon beats its wings. Each creature within 10 ft. of the dragon must succeed on a DC 22 Dexterity saving throw or take 15 (2d6 + 8) bludgeoning damage and be knocked prone. The dragon can then fly up to half its flying speed.
Legendary Actions can be anything. A high-level Legendary spellcaster could fire off four spells per round. A Legendary demon lord could teleport around the battlefield, striking PCs who thought they were safe. A Legendary archdruid could rapidly switch between different beasts, each one making a different attack.
One more thing - if you don't want your boss to fall to the first spell that the control wizard casts, don't forget to add Legendary Resistances. They may feel cheap when you use them, but allowing the boss to auto-save a couple of times helps the encounter be memorable in the right way. Turn 1 banishment against a creature with 8 CHA shouldn't be the answer to everything...
From what I can gather what you're asking, you want to know the probabilistic difference between rolling 10d10 and 5d20. You've rightly pointed out that each roll has the same maximum and that each has a better chance at rolling their given averages. The averages are different, which you already know. They obviously have different minimums (10 vs 5), and so you want to know precisely how different the rolls are.
Dice rolls are commonly noted as "xdy", where x is the number of dice and y is the number of faces. "d" tells us we're looking at dice and acts as a delimiter.
Any time we look at two dice variations where the product of the number of dice (xdy) and the faces of the dice used (xdy) is equal between the variations, we usually want to know how they differ since the ranges are so similar. In our case, 10d10 versus 5d20 is very similar because 10*10 is equal to 5*20. The following answer can be used as an example for any comparison of dice rolls where x and y on both variations have the same product (2d10 versus 1d20, 2d6 versus 1d12, 3d8 versus 4d6 versus 2d12, etc).
Fewer Dice, More Variation
Using AnyDice.com we can calculate the probability very simply with the commands output 10d10 and output 5d20. And really that's all there is to it. The black line below represents 10d10, and the yellow line represents 5d20.
Generally speaking, when you have a greater number of smaller dice, your rolls are less "swingy". Meaning, there are better odds at rolling the "average". But, you have worse odds at rolling higher numbers. When you use fewer number of greater dice, your rolls are more "swingy", meaning you have less chance to roll the average and more chance to roll the extreme ends of the ranges.
Put another way: Look at this graph, it represents the odds that you will roll at least a given number. You can see in general it's better to roll 10d10 because you have greater odds at hitting a certain number until about 60, then 5d20 gives you better odds at hitting those values, but only slightly.
So with 5d20, you have higher odds at hitting a greater range of values, meaning that if you roll 5d20 often, you'll see more "swingy" results. But with 10d10, the odds are more stacked in the middle, meaning it should feel like you're hitting the "average" or the "middle" results more often.
Another Example
But let's simplify. Lets look at output 2d10 vs output 1d20. Same idea as 10d10 vs 5d20. With 2d10, the odds are much different than 1d20 because there are a greater number of rolls that represent the middle values (11). there's 10-1, 9-2, 8-3, 7-4, 6-5, 5-6, 4-7, 3-8, 2-9, and 1-10 representing 11. 10% of all the combinations are 11. But for higher values (20), there is only 10-10 representing that, which is only 1% of all possibilities. But for a 1d20, there is a 5% chance for every number. So 11 is represented by the same number of faces as 20, or 1.
Similarly, if you wanted to compare 1d100 to 5d20 and 10d10, you would see a flat probability: a 1% chance for each value between 1 and 100.
Conclusion, Final Thoughts
So we can see why certain combinations of damage die are used in RPGs, and more specifically D&D 5e (which you originally asked about). The more dice you can use for a given range, the more you, as a designer, can control the probable outcome of that roll. Whereas some rolls, like loot tables, rely on an equal probability of each result by using only 1 (or very few dice) such as 1d100 rolls. Simply put, if you want to design a system that uses dice, you can control probability more by adding more dice.
Best Answer
Mechanically, I would say the main difference is that advantage is easier and provides more stable results (As your result chances go from evenly distributed, to a favorably skewed curve). 5e favors using advantage/disadvantage as a quick, simple, effective way to deal with temporary bonuses.
Narratively, advantage is a temporary bonus to one action, while a change in DC is a change in the difficulty of the task being attempted.
Giving Advantage means the task is harder for Violet to fail
This means, narratively, that thanks to Violet's special effects, the guard will be more likely to be intimidated by the next thing that comes out of Violet's mouth. The bonus is only for Violet's next action (or until the effect that gives Violet advantage no longer applies to the situation).
Changing the DC means the task has changed
This would mean the guard has lost their composure, and they will think everything is more intimidating/scary. In this case, the bonus applies to Violet and to anyone else attempting the same task.
As another example, building a partial bridge over a gap doesn't make jumping over easier; the jump you have to do to get across is now an easier task for everybody.