President Trump noted a couple of things recently regarding our ally, South Korea. That the aircraft carrier Vincent was on its way there. And, that we were looking to China to help rein in North Korea, since they have the leverage and Korea used to be part of China.
Both were incorrect and stupid. And we Americans treated them as the incompetence at play which they were. But it looked different from the point of our allies.
In the first case, the wayward aircraft carrier... For the administration (and most Americans) having the ship go there as promised or a few weeks later is no big deal. For South Korea, it was a major point of contention. To THEM, we were lying and bluffing. Something they see form North Korea all the time. It's never good to be compared to North Korea. And when it turns out that the aircraft carrier was, in fact, heading towards Australia, their media made a big fact of the fact that Russia, China and North Korea were laughing at them because their ally (the US) couldn't be bothered to tell the truth.
As for the China / Korea history, Trump got it way wrong. Now to you and I, this probably doesn't mean much. But (for my cheesehead) it is like the President of China saying that Wisconsin used to bre part of Michigan, and to put pressure on Harley Davidson to lower tariffs, they would talk to the Governor of Michigan.
Korea has been attacked and occupied (rarely) by China, it the two people are very different from their perspectives. It is akin to saying the Italians and Norweigians are the same because they are both from Europe.
Why do we care? Good question class, glad you asked it.
We care because the President of the United States must be seen to be an honest international actor, or his words will be ignored, and that doesn't leave a lot of options. Now, we know the North Korea doesn't like us, but if neither Korea BELIEVES us, then our 40,000+ servicemen on the Korean Peninsula are in danger. As well as the millions of people we have sworn to protect in South Korea.