Sasuke understands, not the generalized stuff Kakashi was using that Sasuke found more alienating than helpful.

Secondly, the Sound Four address Sasuke's curse mark directly.

The curse mark is a handicap for Sasuke. He can't control it (rather it's a constant risk that it will control him), so Sasuke has to live in constant fear of using up too much chakra lest it take over him again. It's a troublesome burden, both mentally and physically.

It's interesting that Kakashi and Sakon both hit Sasuke with a "you're not as unique or special as you thought" moment. But where Kakashi's moment is vague and unclear and pushes Sasuke away, Sakon's reveal that he too possesses the curse mark startles and intrigues Sasuke.

Sakon shows Sasuke that he too has suffered from the curse mark and knows the pain it brings. But what's intriguing about Sakon and the other three Sound ninja is that they can control their curse marks. Sakon doesn't seem scared or alarmed by his curse mark, as he uses it confidently to boost his power and has complete control over it.

This is rather unlike Sasuke, who doesn't know much about controlling his curse mark, what it does or what it is doing to him, nor does Sasuke have anyone in the village who can teach him about the curse mark, because as far as Sasuke is aware, he's the only person in the entire village who suffers the agony of having a curse mark (I don't think he knows about Anko's curse mark). Then suddenly four other kids show up knowing how to control the curse mark with confidence. That has to be an intriguing prospect for Sasuke, since Sakon knows information about the curse mark as well as how it works, helpful information he freely shares with Sasuke.

From the Sound Four, Sasuke learns that using the curse mark will not only risk taking over his mind or sucking up his chakra, but it is eroding his body, and Sasuke is apparently dying.

He's only dying slowly as Sakon bluntly point out, but that tidbit of information would probably impact Sasuke's decision and be terrifyingly alarming. If Sasuke's life is on a clock and he's dying anyway, doesn't it make more sense to abandon the people who can't help you and defect to the people who might be able to fix whatever the curse mark is? Unless I'm misunderstanding something, it sounds like Sasuke is going to die eventually and that presently his life is slowly ticking away. Perhaps Orochimaru can help, because the Sound Ninja don't seem to be dying. Maybe it's just me, but the prospect that I'm dying would freak me out and might cause me to make some desperate decisions I normally wouldn't if I thought there was someone out there who could save me.

Of course there's the possibility that Sakon is lying or bluffing, but there's no way to know for sure, and it's a risky gamble when the stakes are that high. All Sasuke knows is that the curse mark has caused him a lot of agony, and none of the adults in Konoha have addressed the problem except for Kakashi sealing it. But that only works so long as Sasuke's resolve is strong. And considering how much emotional and mental turmoil Sasuke is battling through right now, keeping the mark under control doesn't seem like the easiest thing.

The Sound Four also address Itachi by name, reminding Sasuke not to forget his purpose. Interestingly, the Sound Four never use the word revenge; they use purpose. This lingo speaks to Sasuke because Sasuke sees his goal of killing Itachi in terms of his life purpose, and the Sound quartet appeal to that aspect of his ideology.

Conversely, Kakashi approached Sasuke in a generic sense, treating Sasuke's intention to get revenge like some silly child's game with no meaning or consequences, not like Sasuke's life purpose that Sasuke had centered his life around for the past 6 years.

If the Sound Ninja are more persuasive, it's because they are approaching Sasuke on his terms, most of the conversation is directed on Sasuke: Sasuke's values, Sasuke's curse mark, what they and Orochimaru can do for Sasuke, Sasuke getting stronger, and offering knowledge on how to stop Sasuke dying from the curse mark.

Meanwhile, Kakashi's speech deflected the conversation from Sasuke: he addressed Sasuke's revenge, but also deflected the conversation to what Sasuke needed to do for his team: get a better attitude, stop feeling sorry for himself, protect his comrades, treasure his comrades, feel sorry for his comrades' rough pasts, sacrifice revenge for his comrades.

Whereas the Sound Four concoct their strategy around what they can do for Sasuke: help Sasuke with his curse mark problem, give Sasuke power, help Sasuke pursue his life's purpose/revenge, and they don't mention much about themselves except that they are strong and knowledgeable about the curse mark, but even that they only mention in order to offer Sasuke access to that wealth of strength and knowledge.

In the end, that Sasuke chose revenge over Team 7 probably indicates that Sasuke was leaning ever so slightly more in that direction to begin with. With Sasuke on bad terms with Team 7 (due to his frustration with Naruto and their recent fight), feeling like none of his teammates understood him, and Kakashi demanding that Sasuke give up on his life goal, Sasuke was probably frustrated with his team and tired of feeling misunderstood.

Tayuya's philosophy that, "to gain something, you must also give something up" additionally speaks to Sasuke's own outlook on life. For one, it's true, and additionally this line of thought only reaffirms what Sasuke has already been thinking. Sasuke has been wondering what he's been doing all this time, meaning he feels frustrated with his lack of progress since the Uchiha massacre, like he's been idly messing around, not taking things seriously. Sasuke pins his frustrations on his team, viewing them as a distraction holding him back from diligently training. Tayuya confirms what Sasuke has already been thinking by noting that Sasuke will have to give things up if he wants to gain something else.

The Sound Four played their cards brilliantly to appeal to Sasuke. Down to their very phrasing and careful selection of words, they strategically appealed specifically to Sasuke's outlook and ideology, as opposed to Kakashi, who picked a generic approach that was impersonal and out-of-touch. I suspect Kakashi approached the conversation as if he was talking to a younger version of himself, because Kakashi is probably still under the mistaken delusion that Sasuke is a mini version of himself, not realizing how vastly different Sasuke is from him.