There are 4 variants of each type of dragon, each corresponding to a specific age. At each age, the red and gold dragons each have the same CR (4, 10, 17, and 24). The dragons are superficially similar, with identical AC, hit points, fire breath, legendary actions, and attack bonuses/damage.
But on closer inspection, it is clear that the gold dragon is significantly more powerful than the red dragon: The save DC for their Frightful Presence is higher, they have higher ability scores, they have a couple more skills. But, more importantly, they have 2 abilities that the red dragon does not: Weakening breath and change shape.
Weakening breath is extremely effective at limiting the combat effectiveness any strength based creature. Shapechange would allow an ancient gold dragon to change into any monster in the Monster Manual (except the Tarrasque) and create equipment for that creature and keep its HP and stellar INT, CHA, and WIS scores and keep its legendary resistance. This is crazy powerful! (Anything that can turn into a noble genie at will should be frightening.)
Another point of difference between the two types of dragons is their lair actions. This is the one place where I would say that the red dragon has an advantage over the gold: The red dragon has a lair action that can cause damage, while the gold dragon does not. (Although that action requires line of sight for the red dragon.) The remaining lair actions are analogous to each other. Both have an action to grant the dragon advantage on attack rolls, and one to waste enemy actions. The gold dragon's future-seeing is generally better than the red dragon's tremor because it is guaranteed to work and applies to ability checks and saving throws as well, while the red dragon's volcanic gasses is generally more powerful than the gold dragon's banish even though it targets a stronger save because it effects an area, breaks concentration, and doesn't allow a second chance to break.
The red dragon's lair actions are a bit stronger than the gold's, but how on earth are these two monsters the same CR when one of them has access to the entire Monster Manual?
Best Answer
CR ignores versatility
I will preface addressing the two "extra" abilities the Gold Dragon has over the Red Dragon by saying that the CR calculation described in the DMG ascribes no value whatsoever to versatility. The main way that you calculate the effect a monster's battlefield impact has on its CR is its damage:
Thus if I give a monster 2 attack options that allow that monster to deal damage in slightly different ways, each of which will be useful in a specific situation, but overall they are equally effective, the monster would have the same CR as if it only had one of those options.
The DMG does provide a table of special traits that affect CR, and suggests this should be extended for custom traits - but those only ever affect CR if they increase the capabilities of the creature without opportunity cost, not if they simply increase a monster's options.
Another example of this is the half-dragon template from the Monster Manual:
Here there's a specific CR under which adding the Breath Weapon to a creature doesn't matter, because it's assumed its other attack options will deal as much or more damage, and giving the creature another option doesn't change the CR.
So - CR ignores versatility, and only cares about raw power.
That being the case, I will now demonstrate that these two features are side-grades to the rest of the dragon's powerset - and thus that CR ignores them. Beyond that, I do genuinely think the majority of the time in a fight, the dragon will be better off not using them.
Change Shape is almost always a downgrade in combat
The dragon can turn into humanoids or beasts only; no weird aberrations that have absurdly powerful situational abilities, and no "noble genies".
The highest CR published Beast is CR 8 (T-Rex). Prior to the publication of Mordenkainen's Tome of Foes (i.e. when the Adult Gold Dragon was published), the highest CR humanoid was CR 12 - now there are a few above it.
As such, the dragon will be using this ability to turn into things less powerful than it is, since only the CR 17 and 24 versions of these dragons have this ability; yes it may gain some situational powers, but as previously stated CR ignores versatility.
It also retains its HP - so unlike other polymorph-like abilities, it cannot use this to soak damage.
Finally, there is nothing in the ability that would suggest it forms the equipment of the creature it changes into.
Weakening Breath is usually not that good compared to Fire Breath
The gold dragon must choose, each time it has an opportunity to use its breath weapons, which one it wants to use.
The Adult Gold Dragon, for instance, can choose to do one of the following things in a massive area:
Weakening Breath has a big "weakness"; If the creatures being targeted are a threat because of their Strength-based attack rolls, they are likely to have high Strength and perhaps Strength saving throw proficiency. Target a high level Strength-based Fighter with this, and they will have about half chance to succeed. The ability redeems itself a little by causing Strength saving throw disadvantage, thus making it harder to throw off once it sticks, but it has to connect first.
By using this option, you're giving up almost certainly about 66 damage on this creature, since if it's strong in Strength it's likely to be weaker in Dexterity saves. And that is almost certainly half its HP or more.
As well of that, it's important to remember that this is an area attack; Dexterity-based creatures or spellcasters in the area won't care about Strength disadvantage; but they will definitely care about damage - likely more than the Strength-based creature.
Sometimes Weakening Breath will be useful - but most of the time Fire Breath is better. And CR doesn't take this into account because again, it ignores versatility.