If a creature doesn't need to breathe that means it (basically) never needs to breathe; I found no features where holding your breath is helpful, only features where not needing to breathe is helpful
The cloudkill spell shows us that "not needing to breathe" and "holding your breath" are indeed different things, otherwise there would be no reason to list both.
Not needing to breathe is not defined by the rules so we would use the standard English meaning of "not needing to breathe" which is that something never needs to breathe.
If I say that a fish needs water to survive it doesn't mean that the fish needs water right now it means that they always need water.
Similarly, if I say that a human doesn't need cobalt to breathe it doesn't mean that they don't need cobalt right now it means that they never need cobalt to breathe. (Extreme situations where somehow cobalt is required for breathing could exist, but this just shows that "never needs cobalt" isn't really a phrase that works; it really means "in all likely scenarios, won't need cobalt")
I found only two mentions of the rules where breathing has negative effects:
High Altitude (DMG page 110):
Traveling at altitudes of 10,000 feet or higher above sea level is taxing for a creature that needs to breathe, because of the reduced amount of oxygen in the air. Each hour such a creature spends traveling at high altitude counts as 2 hours for the purpose of determining how long that creature can travel.
A creature can hold its breath for a number of minutes equal to 1 + its Constitution modifier (minimum of 30 seconds).
When a creature runs out of breath or is choking, it can survive for a number of rounds equal to its Constitution modifier (minimum of 1 round).
Holding your breath is arguably "helpful" here but it's also effectively not a choice. You would be holding your breath unless you specifically chose not to. Additionally you can only suffocate if you actually need to breathe in the first place. Not needing to breathe would thus protect you from anything which causes you to suffocate, such as quicksand (DMG page 110).
Unfortunately, neither of these are removed by holding your breath, as you would still be a creature that needs to breathe.
Even Inhaled Poisons still affect a creature holding their breath as it says (DMG page 257):
Holding one's breath is ineffective against inhaled poisons, as they affect nasal membranes, tear ducts, and other parts of the body.
There is also an item from Volo's Guide to Monsters; the Survival Mantle (page 81) which states:
A creature wearing a survival mantle can breathe normally in any environment (including a vacuum) and has advantage on saving throws against harmful gases (such as those created by a cloudkill spell, a stinking cloud spell, inhaled poisons, and the breath weapons of some dragons).
This seems to list the things which relate to breathing; however, I've now checked the listed spells, inhaled poisons, and all breath weapons that dragons get. None of these have any text saying that holding your breath will do anything to counter them.
Regarding the clarifying text at the end of each spell:
A spell affects anything that isn't explicitly stated as not being affected. The cloudkill spell likely calls out that it does effect creatures that can't breathe simply because it would be a common point of confusion, and a disconnect from our real-world logic, if a gas was somehow affecting creatures which can't even breathe in said gas.