The answer is NO, you have to check with each country. As this E.U. law have to be signed by every country and it's far from being done everywhere (IE UK do not signed it, so it's not applicable there). Check country per country, and be careful not to cross a border without knowing the situation on the other country.
Examples: For France you have to get a Schengen visa, the only advantage is that it's free (no fee). For going to UK, my Wife needs to get an EEA Family Permit from the British Embassy to be able to travel there.
I remember a story of a Filipina having a Schengen visa, they were going by car from one Schengen country to another but they had to pass thru an European non Schengen country (the motorway was going through this one). She was denied entry at the border on the next Schengen country because her Schengen visa was one entry only. A lot of mess in their holidays has they can't travel any more to a Schengen country...