The problem may actually be that, although you think the attributes of the original data and the new domain values are the same, the domain values most likely use numeric values or short text codes internally, and only show you the longer, human readable, label or description as a more user friendly substitute.
This means you need to convert your original data to the appropriate numeric or short text code values as you defined them yourself during domain creation (under "Coded Values" / "Code" and "Description" in the dialog), to have them show up with the correct description in the tables.
E.g.:
If you had an original field named "City" with already entered values "New York" and "Chicago", and you created a domain like:
"Code", "Description"
1, New York
2, Chicago
and you subsequently associated this domain with the field "City", than the original text values "New York" and "Chicago" wouldn't display, since the allowed values for the field are 1 and 2, not "New York" and "Chicago", from the moment you associate this domain with this field.
If you changed the original values to 1 and 2 respectively, the attribute table would show the domain descriptions "New York" and "Chicago".