Does a cryptosystem provide unconditional security if and only if it provides perfect secrecy?
Yes, in context of encryption the two terms have the same meaning, i.e. even an unbounded adversary cannot succeed in breaking the secrecy of the scheme. 

Note that there may be other primitives which also provide unconditional security with respect to some other notions, such as information theoretically secure message authentication codes with respect to unforgeability.