It seems that the term “Continuum of Care” came from the health care industry in reference to major illnesses such as cancer, where as the term “Wrap Around Services” was developed in the psychiatric field, namely for children with mental health issues and their families.
A difference could be made by saying that Continuum of Care consists of housing and wrap around services, and that Wrap Around Services could be the designation for everything done for the homeless person outside of housing. But again, it does not seem that homeless services do not make such a clear distinction.
Because so many different professions and different government departments have worked on the homeless issue independently of each other, they have inevitably developed different jargon and definitions. Homelessness care is still in its early stages of development so it will be a while before the language of the work is honed down to single specific terms.