There is a hypothesis called the "Critical Age Hypothesis". It postulates that learning a language becomes much harder with the onset of puberty because there are physical changes in our brains - particularly the region that controls language - after the age of 12. This doesn't mean that we're not capable of learning a language at all but the hypothesis is that we will never acquire native-like competency in a language if we only start learning it after the onset of puberty.
There are also several other theories in Second Language Acquisition studies that show that adults self-monitor a lot more than children, which thereby hinders their progress in a language. We also reach a plateau a lot earlier than children and, the older we are when learn a language, the sooner we reach this plateau. The plateau is different for every person though, which is why some people seem to find language learning easier than others.
A third point to consider is the motivation for learning the target language. If a person is internally motivated to learn a language - ex. (s)he wants to be able to communicate with people socially to avoid isolation or exclusion - language acquisition is likely to be more successful and the person will seem to advance more quickly. If the motivation is external, such as needing English for a promotion, the progress and acquisition will be much slower.
Yes, it is entirely possible that many of your students are genuinely studying a lot but not making much visible progress in your opinion. I have spent hours studying Korean but my limited understanding and speaking abilities don't reflect this. Some adults will have a sub-conscious mental block too. The amount of study time doesn't reflect how successful their acquisition is because it depends on what they're doing in that study time. If they're just memorising grammar rules and vocab, they're not going to be very successful.
With regard to vocab acquisition, we have productive and receptive vocabularies. This means that the vocab that we receive doesn't automatically become the vocab that we can produce. Think about your own Korean studies: How many times have you had to have a word repeated and then used before you start to remember it? This differs for everyone. Studying vocab is one thing, using it is completely different. If your students are studying a lot of vocab that they never really get a chance to use, they're going to take much longer to learn those words.
Even more difficult in English is that we rely a lot on nuances when we speak. We also use a lot of inflections, which makes speaking a lot harder for Koreans in particular because this is a crucial difference between Korean and English. Syllable emphasis is hard for them, and something that we, as mother tongue speakers, do naturally. You also have to allow for transference from the first language. It is common for us to impose language structures from our first language onto a target language - i.e. direct translation - and this is one of the most difficult things to change as an adult learner.
Be careful with how you correct your students too. Adults are more inclined to "shut down" if they feel that they are not making enough progress or if they feel that they make too many mistakes. Don't be too direct in your corrections although it is sometimes necessary. If it is a small mistake like a pronoun reference, do a "model correct" by emphasising the correction with a stronger accent when you say that particular word. This is especially useful when working with higher level students. For the lower level students, focus on communicating meaning first. If the student can get their point across, that's great even if the grammar isn't always right. Then focus on fluency: Can they speak without stopping every few seconds even if the grammar is incorrect. When they can communicate their meaning in a fairly fluent manner, start to focus on the grammar structures and work your way down to the smaller errors that distinguish them from first language speakers.
Finally, remember that there are many different approaches to learning languages. Some people prefer the grammar translation approach (which usually means that they can't speak the target language well but they can read, write and understand it) which is pretty common in Korea. If your students are focused on memorising grammar rules and vocab, they are probably using this approach.
Language is not stagnant. It must be used in order to be solidified. This means that if a student doesn't try to use the language (especially in speaking) even if they produce imperfect utterances, their language acquisition will be very, very slow. They need to expose themselves to the language on a daily basis and for short periods of time. For a lower level student, suggest that they read one short (300 - 500 words) news article every morning (suggest sites for them), listen to a short video clip (1 - 2 minutes max.) daily and try to speak English for 5 minutes a day (this may be the hardest part for them because they won't necessarily have someone with whom to speak English). Increase the lengths as they improve. If a person is not exposed to the target language on a regular basis, (s)he won't acquire it as quickly. You are surrounded by Korean everywhere you go, which is why you are acquiring Korean more quickly than some of your students.
Adults who are learning a language are usually more motivated. HOWEVER, their progress can be a lot slower and they are more easily discouraged than younger students. Be patient with them. Don't focus on speeding up their rate of learning because there is no standard for this with adult learners. Adults tend to understand concepts fairly quickly but take a lot longer to implement them whereas it's the complete opposite with most children. They may be making the same mistakes repeatedly when they speak but are most likely aware of the grammar rule/concept behind the error. Instead of correcting the error, ask them if they remember the rule for that particular error. That's when you'll probably see the results of their studying.
Unfortunately, the older your students are, the more likely they are to have reached their plateau. This means that no matter how much they study, they are not going to improve their English any further. However, if they stop using the English that they do have, it will fossilise (depending on how much they had acquired already and their level of proficiency) or disappear entirely.
Listening and reading are not enough to learn a language; it has to be spoken too and that, for at least 90% of the adults I've taught and studied (including myself), is the hardest part because most of us feel self-conscious when we make an error, can't make ourselves understand or, in a classroom environment, compare ourselves to people with a higher proficiency in the target language.
