Are phone numbers best saved with country calling code (and/or plus) or not?
With fixed location landlines if we wanted to dial a number in another country, we dial the “exit code” first.  In the United States the Exit Code is 011 and is often referred to as the international dialing prefix.  Dialing 011 told the phone system to expect a country code next.  

Dialing patterns and parsing had to updated with mobile phones.  The + symbol became the international prefix that stands for ‘what follows is a complete dialing instruction including country code’ and no further parsing attempts are necessary. 

But why not just country code? Why do we need the plus(+)?  If 1 is the United States country code isn’t that enough?  Not really, let’s say I’m visiting in Mexico and have my friend in the US 7 digit number stored as 523-xxxx (which is parsed correctly IF I’m at home in the US AND within my local area code) and I try to call that.  Is the 52 at the start of the number Mexico’s country code 52 or is it something else?  If we take pop culture into account is 867-5309 a 7 digit US number or is it a short code in China (country code 86)?  Dialing from a landline it’s treated as a local number because we didn’t dial the exit code 011 first.  

Without a plus the carrier has to further apply parsing rules, how many digits were dialed?  Are there digit combinations that mean something?  If it was 7 digit could this be a local, if it’s 10 digit it might be an area code and phone number in the US, or it might not be.  If its 11 digits starting with 1 it’s probably a US number.  But in some countries local numbers might start with a 1, in fact some counties use 111 as an emergency number.  

For most compatibility always add the + prefix.  It doesn’t matter where you are (I.e. which carrier is hosting your connection) it’s understood that what follows the + is a compete dialing instruction including country code, area code (if applicable), and local number.  (1 is the country code for United States)