I completely agree with Eun,she's got a good point.
To add some more to Eun's good reply,
we call that kind of words "adopted words" which was borrowed from other languages like Chinese or English, sometimes Japanese.
As you know, the word "미팅(meeting)" was borrowed from English,
and "회의(會議)" came from Chinese,I think.
(Actually "회의" is "会议(후이-이)" in Chinese, and "会議(かいぎ) in Japanese)
Also they have the same meaning at least in Korea,but "미팅" is less formal than "회의".
To sum up,
1) "미팅" has two meanings,
One is "a blind date", and
another is "a business meeting"
2) "회의" is only for "a business meeting"
3) "회담(會談)" is also used to mean "a meeting", but it's very formal.
For example,
"the G20 summit conference" - "G20 정상회담"
^_____^