That's not what the rule book says. The rule book says being invisible means you can always "try to hide", not that you are without need to try or that your tries don't normally involve a Dexterity (Stealth) check.
That is true. The rule does say you can always try to hide while invisible. If an invisible rogue were rubbing at someone and shouting at their face, I wouldn't bother to let the rogue roll the stealth check, though, even if the rules clearly state, as they do, that the rogue can. I would just tell the player the attempt failed.
By reading the hiding rules it also seems to imply that it is not possible to try to hide when there is no cover available and the area is clearly lit. No sneaking past a guard in a corridor by trying to move past his back, as there is no rules for facing, etc. I rule over these limitations and allow the PCs to do whatever makes sense.
One of my players, after finding an elven royal hunting regalia, that consisted of, among other items, the cloak and boots of elvenkind, read the rules for both and asked me right away why would he ever bother to wear the boots, as the rules part of the description of the cloak would generously give advantage on all stealth checks. I told him I would not run it exactly that way. Instead, I would do what seemed to me the intent of that line, that was to let the cloak give advantages on stealth tests related to
sight, the same way the cloak would impose disadvantage on perception tests related to sight. I is definitely
not what is written there, I cannot and will not dispute it. But it is one (not necessarily the only) logical way to apply the rules, that clearly solves any kind of otherwise unexpected interactions.
Tests related to sight? Cloak gives disadv. to the observer and adv. to the user. Invisibility makes attempts to see straightforward fail.
Tests related to listening? Cloak and invisibility don't help. The boots give advantage.
How to put all this together? Assessing the situation. Sneaking past a guard in a silent dungeon? The guard perception will be based on whichever is better for him, between sight and hearing (assuming the sneaking PC is not stinking or bumping at the guard). So invisibility rules out sight, but hearing is unaffected. Cloak gives advantage against sight, but hearing again is unaffected. Boots give advantage against hearing, but now sight is unhampered. Cloak + boots or invisibility + boots will therefore result in advantage for the sneaking side, as the guard cannot rely on any unhampered sense. I tend to consider, for humans specially, sight as the more reliable sense. If the sneaking character was too far away (for instance trying to sneak over a field), or the room was noisy, I would rule that were the guard to rely on hearing, he would have automatic disadvantage regardless of sneaking PC magical item worn, or even fail the test straight away (I don't think a human can listen to a cat walking 100 m away. A dog might, though).
All in all, the above might seem long and complicated, but it really isn't. I can make all the above assessment instinctively during play. It could be ruled differently. I could allow the cloak magic to actually enhance all the potential sneaking talent of a PC, not just camouflage it. This would be surely closer to the letter of the rule, but some would, as I did, find it farther from the heart of it. I didn't want the "stronger" interpretation because I find it cool that the cloak and boots benefit of working together, allowing the PC to be better at sneaking on broader circumstances, but without resulting in number inflation. Invisibility, according to my rulings is simply better than the cloak, but looking on the magic items list, the invisibility ones are rarer and more limited in use, so I think the ruling is fair.
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