They are both a room at schools where classes are taught.
It is one of the things called differently in college than in grade or high school.
For example:
class: 수업 (up to high school) vs 강의 (college)
classroom: 교실(classroom) vs 강의실(lecture room - college)
start of semester: 개학 vs 개강
end of semester; 방학 vs 종강