it's confusing because they can mean same or differently depending on context. speaking of difference, 늘다 is used when saying something is getting better. ex) 축구 실력이 늘었다, 한국어 실력이 늘었다. 늘어나다 is uesd when saying length or height is extended. ex) 고무줄을 당기면 늘어난다, 스프링이 늘어난다. 증가하다 is used when saying amount,figure or number of something is increased. ex) 여행객의 수가 증가했다 or 여행객의 수가 늘어났다 can be used as well.

늘다 always\늘어나다 increase\증가하다 increase