In terms of the United States, there is a distinction, at least in my understanding as a former US Marine.
The United States Armed Forces are the Navy, Army, Marine Corps, Air Force, and Coast Guard.
The Unites States Senate Committee on Armed Services encompasses Legislative oversight of the Department of Defense (which oversees most of the US Armed Forces), but other areas of military interest or national security.
Therefore, I would suggest that, in terms of moving from more general to more specific, the order of the three phrases would be:
- Armed Services (all organizations dealing with military interests or national security)
- Military (all organizations dealing with military interests)
- Armed Forces (the five organizations: Navy, Army, Marines, Air Force, Coast Guard)
So, I was a US Marine, part of the Department of the Navy, and therefore was a member of the Armed Forces. As a member of the Armed Forces, I was part of the US military. As a member of the US military, I was also a member of the Armed Services.
From the other direction, a TSA agent could be considered part of the Armed Services of the US, but is not military, and is certainly not a member of the Armed Forces.
Cross-over occurs with the Coast Guard, which is part of the Armed Forces, but not part of the Department of Defense (they are under the Department of Homeland Security, which also oversees TSA).
The word "military" probably gets misused the most, since it also can be an adjective, as in "military action," "military assets," "military interests," and so on. In those usages, it is very generic, so it's not surprising that "military" as a noun also gets used more generically than perhaps it should.