The United States' country code (1) is not the same as the digit 1 that you are required to dial before the area code and local phone number to make a long distance call from within the United States. It is a coincidence that they are the same number, but +1 603 555 1234 and 1 603 555 1234 have very different meanings, as shown below.
Long Distance Dialing
+1 603 555 1234 - This is a fully-qualified number that includes the country code of the number (in this case, the country code of the United States (1), since the number is in the US). This is the number that you would dial from a country outside of the United States to reach the 603 555 1234 phone number (replacing the plus sign (+) with the international access code of the country from which you are calling).
1 603 555 1234 - This is what you would dial to reach the 603 555 1234 phone number when calling within the United States, assuming that calling to the 603 area code is a long distance call from your current location. From within the 603 area code, you would only have to dial 555-1234.
See also
Modifying a Normalization Rule
Set up dialing and normalization rules for a server in the United Kingdom (UK)