Conference, n.
Etymology: [F. conférence. See Confer.]
1. The act of comparing two or more things together; comparison. [Obs.] Helps and furtherances which . . . the mutual conference of all men's collections and observations may afford. Hocker.
2. The act of consulting together formally; serious conversation or discussion; interchange of views. Nor with such free and friendly conference As he hath used of old. Shak.
3. A meeting for consultation, discussion, or an interchange of opinions.
4. A meeting of the two branches of a legislature, by their committees, to adjust between them.
3 more senses
5. (Methodist Church)
A stated meeting of preachers and others, invested with authority to take cognizance of ecclesiastical matters.
6. A voluntary association of Congregational churches of a district; the district in which such churches are. Conference meeting, a meeting for conference. Specifically, a meeting conducted (usually) by laymen, for conference and prayer. [U. S.] -- Conference room, a room for conference and prayer, and for the pastor's less formal addresses. [U. S.]
Shades of meaning
Fernald has no entry of its own for conference; he lists it under company, conversation.
Where he writes about it:
A conference is more formal than a conversation. — from conversation