1. "Gross federal debt" includes the money which Congress borrowed from (future) Social Security recipients in order to pay for (current) tax-cuts and middle-east invasions and whatnot.
2. "Held by the public" excludes that portion, focusing on money borrowed from individuals/companies/other-countries that voluntarily lent their money to the US government.
So we should make our decisions based on the first one, unless someone is plotting to screw millions of American taxpayers out of the money they already paid as insurance-premiums. :p
[0]: https://fred.stlouisfed.org/series/FYGFGDQ188S