The DXS was a fictionalised version of the CIA. It was only really interested in overseas operations (the CIA isn't even supposed to operate inside US borders).
The Phoenix Foundation was pure fiction; I don't know of any real-life organisation that parallels it. Pity!
Other important distinctions:
The Phoenix Foundation often worked with or for the US government; the DXS
was part of the US government. This put Mac in a very different legal situation in his work.
The DXS was focused on intelligence operations; the Phoenix Foundations was focused on science and research.
The Phoenix Foundation did not officially engage in any covert operations. Of course, the DXS didn't "officially" do so either.
The Phoenix Foundation, not being a governmental body, could serve as a neutral party or as presumably disinterested observers or mediators (
Cease Fire,
Humanity,
The Wall, etc.). It could also act as an advocate for groups within the US (
Trail of Tears).
The Phoenix Foundation would not have been viewed with the same level of suspicion as the US government. This is mentioned quite a few times, although often obliquely.
Phoenix was able to solidly retain its own liberal attitudes, without being in any danger from the changing politics of US governmental administrations.
As Lothi said, Phoenix was privately funded; the DXS would have been paid for with taxpayer money. Mac's ultimate boss at the DXS would have been the US President. His ultimate boss at Phoenix would have been the Foundation's Executive Director, who in turn was answerable to the Chairman (chairperson) of the Board of Directors.
Pretty much all of Mac's DXS work was highly classified. Most of his Phoenix work was not.
Mac's work for the DXS consisted entirely of specific, short-term assignments. His work for Phoenix often involved long-term assignments and research projects; episodes often started with him in the middle of an assignment that had been going on for a while.
Mac worked part-time, in a largely unofficial capacity, for the DXS; he also did jobs for other departments or organisations. He seems to have worked on a less ad-hoc basis for the Phoenix Foundation. In both cases, he had a lot of leeway over whether he would take on any given project or job.
'Beth