Data formats matter! NOTE that the following info is as current as I can make it as of the end of March 2018. If you see a mistake here, please let me know!
I TRY TO USE OFFICIAL FORMATS EVERYWHERE
In all areas of the work that I do I attempt to format the data clients enter into my databases in a manner that complies with current recognized standards. So for example, I try to format mailing or street addresses according to the recommendations of the USPS in publication 28 (no commas or other extraneous punctuation, no extra spaces, etc).
THE NANP AND PHONE NUMBER FORMATS
This is a bit unclear but the current "rules" for phone numbers seem to be set by the North American Numbering Plan (NANP), which prescribes the format 713-555-0123.
Back when seven-digit dialing was still the rule in many parts of the country, the use of parentheses for the area code was intended to suggest what parentheses usually suggest, that is, that something is sort of optional. But that old system has changed in two major ways.
So to the extent that there is such a thing, the "official" rule for phone numbers, set by the association that promulgates rules about this, is to format them 713-555-0123. Formatting them per this rule means that the numbers are less likely to cause compatibility problems with other systems (say, if you export contacts from a database and import into some other application or data-management system) and less likely to cause a hiccup when the phone is passed as a parameter to a dialing system.
BUT THE RULE DEFINITELY AIN'T A LAW
And it's worth noting that, while the NANP's rule is followed by the government and widely respected by programmers, much of the world is simply unaware of it — or is aware of it and doesn't care.
That said, the format used for the display of phone numbers is just packaging. Before a phone number is used to dial the phone, it always get stripped down to a simple string of numbers. I suspect that the most important issue, from the user's perspective, is consistency, rather than "correctness".