As mentioned by SSD and kviiri, there is some mathematical difference, mainly in that Advantage is consistent and you don't need to calculate by how much you will change the DC.
Tezra also gives a good answer on the interpretation of it: Advantage means the character has some help, something that makes the task easier. A character climbing with a rope has advantage. Climbing something with no inclination has a small/no DC.
I will disagree with the two most upvoted answers, though, in that your scenario seems to be more about the Guard getting more scared of anything than of specifically Violet - but that is my interpretation of how I imagine the guard reacting1. Although mechanically advantage is easier, straight-forward and the standard way in 5e, this specific scenario might call for a reduction of DC.
A similar case happens in Lost Mine of Phandelver, where feeding the hungry and angry wolves will make the DC to calm them down lower, not only give advantage to the person who fed them. That's to say there is precedent in the published adventures for reducing the DC.
A character who tries to calm the animals can attempt a DC 15 Wisdom (Animal Handling) check. On a success, the wolves allow the character to move throughout the room. If the wolves are given food, the DC drops to 10.
Also, there is one mechanical difference that seems to not be mentioned until now: Advantages do not stack, additive bonus do - so, if you give a -5 to the DC, you can later on, because of some other action, give another -5 to the DC. If you give advantage, you will not be able to give another advantage later on. Note that this is intended to reduce the bean counting, in opposition to what happened in earlier editions.
Either way, a good rule of thumb, as mentioned by Tezra, is: If the task became easier (or harder) to everyone, or if you essentially changed the task, the DC changes. If someone has some kind of particularity that will help them specifically, give them advantage. In your scenario, say the guard is scared of Undeads and Violet disguises herself as a Zombie. That would be a clear advantage case.
1 To be clear, I'm imagining the guard becoming kinda paranoid and jumping scared even if a cat meow'd at him after all of this "supernatural" stuff, mostly because I'm imagining the guard does not know/understand that the bard is the one causing all this stuff and is pinning it on some supernatural stuff, like a cliche horror movie.