Contact is the broad arm responsible for exploration and relations with other civilisations; Special Circumstances is its smaller, sharper division — the part that does the morally difficult, deniable work of intervention and espionage.
Minds are central to both: they plan, recruit, and manipulate, and carry much of the burden of deciding when a utopia is entitled to meddle in the affairs of those who never asked for its help. Much of the drama of the series lives in exactly that tension. See also The Culture.